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Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces

Ahmet Eroglu

Author Affiliations

Department of Mathematics, Nigde University, Nigde, Turkey

Boundary Value Problems 2013, 2013:70  doi:10.1186/1687-2770-2013-70

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/70


Received:25 July 2012
Accepted:17 March 2013
Published:2 April 2013

© 2013 Eroglu; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, it is proved that both oscillatory integral operators and fractional oscillatory integral operators are bounded on generalized Morrey spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M1">View MathML</a>. The corresponding commutators generated by BMO functions are also considered.

MSC: 42B20, 42B25, 42B35.

Keywords:
generalized Morrey space; oscillatory integral; commutator; BMO spaces

1 Introduction and main results

The classical Morrey spaces, were introduced by Morrey [1] in 1938, have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces.

Morrey spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M2">View MathML</a> are defined as the set of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M3">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M4">View MathML</a>

Under this definition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M2">View MathML</a> becomes a Banach space; for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M6">View MathML</a>, it coincides with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M8">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M9">View MathML</a>.

We also denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M10">View MathML</a> the weak Morrey space of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M11">View MathML</a> for which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M12">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M13">View MathML</a> denotes the weak <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M14">View MathML</a>-space.

Definition 1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M15">View MathML</a> be a positive measurable function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M17">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M18">View MathML</a> the generalized Morrey space, the space of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M19">View MathML</a> with finite quasinorm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M20">View MathML</a>

Also, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M21">View MathML</a>, we denote the weak generalized Morrey space of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M11">View MathML</a> for which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M23">View MathML</a>

According to this definition, we recover the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M25">View MathML</a> under the choice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M26">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M27">View MathML</a>

The theory of boundedness of classical operators of the real analysis, such as the maximal operator, fractional maximal operator, Riesz potential and the singular integral operators etc., from one weighted Lebesgue space to another one is well studied. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M28">View MathML</a>. The fractional maximal operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M29">View MathML</a> and the Riesz potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M30">View MathML</a> are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M31">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M32">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M33">View MathML</a> is the Hardy-Littlewood maximal operator. In [2], Chiarenza and Frasca obtained the boundedness of M on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M34">View MathML</a>. In [3], Adams established the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M30">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M34">View MathML</a>.

Here and subsequently, C will denote a positive constant which may vary from line to line but will remain independent of the relevant quantities.

The Calderón-Zygmund singular integral operator is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M37">View MathML</a>

(1.1)

where K is a Calderón-Zygmund kernel (CZK). We say a kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M38">View MathML</a> is a CZK if it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M39">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M40">View MathML</a>

(1.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M41">View MathML</a>

(1.4)

for all a, b with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M42">View MathML</a>. Chiarenza and Frasca [2] showed the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M43">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M34">View MathML</a>.

It is worth pointing out that the kernel in (1.1) is convolution kernel. However, there were many kinds of operators with non-convolution kernels, such as Fourier transform and Radon transform [4], which both are versions of oscillatory integrals. The object we consider in this paper is a class of oscillatory integrals due to Ricci and Stein [5]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M45">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M46">View MathML</a> is a real valued polynomial defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M47">View MathML</a>, and K is a CZK.

It is well known that the oscillatory factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M48">View MathML</a> makes it impossible to establish the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M14">View MathML</a> norm inequalities of (1.5) by the method as in the case of Calderón-Zygmund operators or fractional integrals. In [6], Chanillo and Christ established the weak <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M50">View MathML</a> type estimate of T.

A distribution kernel K is called a standard Calderón-Zygmund kernel (SCZK) if it satisfies the following hypotheses:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M51">View MathML</a>

(1.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M52">View MathML</a>

(1.7)

The corresponding Calderón-Zygmund integral operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53">View MathML</a> and oscillatory integral operator S are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M54">View MathML</a>

(1.8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M55">View MathML</a>

(1.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M46">View MathML</a> is a real valued polynomial defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M47">View MathML</a>. In [7], Lu and Zhang proved that S is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M14">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M59">View MathML</a>. In [5], Ricci and Stein also introduced the standard fractional Calderón-Zygmund kernel (SFCZK) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M60">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M61">View MathML</a>, where the conditions (1.6) and (1.7) were replaced by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M62">View MathML</a>

(1.10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M63">View MathML</a>

(1.11)

The corresponding fractional oscillatory integral operator is defined by (see [8])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M64">View MathML</a>

(1.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M46">View MathML</a> is also a real valued polynomial defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M47">View MathML</a>. Obviously, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M69">View MathML</a>. Partly motivated by the idea from [9,10] and the results of [11], we now give the results of this paper in the following.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M70">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71">View MathML</a>satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M72">View MathML</a>

(1.13)

whereCdoes not depend onxandt. IfKis a SCZK and the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53">View MathML</a>is of type<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75">View MathML</a>and any polynomial<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76">View MathML</a>the operatorSis bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M78">View MathML</a>.

Moreover, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79">View MathML</a>andKis a CZK operator, the operatorTis bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M81">View MathML</a>.

Theorem 1.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76">View MathML</a>is a polynomial, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71">View MathML</a>satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M87">View MathML</a>

(1.14)

whereCdoes not depend onxandt. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M88">View MathML</a>the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M89">View MathML</a>is bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M90">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M91">View MathML</a>and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79">View MathML</a>the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M89">View MathML</a>is bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M95">View MathML</a>.

For a locally integrable function b, the commutator operator formed by S (or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M89">View MathML</a>) and b are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M97">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M98">View MathML</a>

Theorem 1.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71">View MathML</a>satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M102">View MathML</a>

(1.15)

whereCdoes not depend onxandt. IfKis a SCZK and the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53">View MathML</a>is of type<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74">View MathML</a>, then for any polynomial<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76">View MathML</a>the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M106">View MathML</a>is bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M78">View MathML</a>.

Theorem 1.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76">View MathML</a>is a polynomial, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71">View MathML</a>satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M115">View MathML</a>

(1.16)

whereCdoes not depend onxandt. Then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M116">View MathML</a>is bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M91">View MathML</a>.

2 Some known results in generalized Morrey spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M119">View MathML</a>

In [9,10,12,13] and [14], there were obtained sufficient conditions on weights <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M121">View MathML</a> for the boundedness of the singular operator T from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M122">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M123">View MathML</a>.

The following statements were proved by Nakai [14].

Theorem ALet<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M17">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M15">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M126">View MathML</a>

(2.1)

whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M127">View MathML</a>, wherec (≥1) does not depend ont, rand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M128">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M129">View MathML</a>

(2.2)

whereCdoes not depend onxandr. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M88">View MathML</a>the operatorsMandTare bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M131">View MathML</a>and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79">View MathML</a>, MandTare bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M133">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M134">View MathML</a>.

Theorem BLet<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M84">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M138">View MathML</a>satisfy the conditions (2.1) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M139">View MathML</a>

(2.3)

whereCdoes not depend onxandr. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M88">View MathML</a>, the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M142">View MathML</a>are bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M131">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M144">View MathML</a>and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M142">View MathML</a>are bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M133">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M149">View MathML</a>.

The following statements, containing Nakai results obtained in [13,14] was proved by Guliyev in [9,10] (see also [15,16]).

Theorem CLet<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M150">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71">View MathML</a>satisfy the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M152">View MathML</a>

(2.4)

whereCdoes not depend onxandt. Then the operatorsMandTare bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M78">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M155">View MathML</a>and from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M81">View MathML</a>.

Theorem DLet<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M160">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71">View MathML</a>satisfy the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M162">View MathML</a>

(2.5)

whereCdoes not depend onxandr. Then the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M142">View MathML</a>are bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M91">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M155">View MathML</a>and from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M95">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79">View MathML</a>.

The following statements, containing Guliyev results obtained in [9,10] was proved by Guliyev et al. in [11,12].

Theorem ELet<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M150">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71">View MathML</a>satisfy the condition (2.4). Then the operatorsMandTare bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M78">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M155">View MathML</a>and from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M81">View MathML</a>.

Theorem FLet<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M160">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M71">View MathML</a>satisfy the condition (1.14). Then the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M142">View MathML</a>are bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M77">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M91">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M155">View MathML</a>and from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M80">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M95">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79">View MathML</a>.

Note that integral conditions of type (2.3) after the paper [17] of 1956 are often referred to as Bary-Stechkin or Zygmund-Bary-Stechkin conditions; see also [18]. The classes of almost monotonic functions satisfying such integral conditions were later studied in a number of papers, see [19-21] and references therein, where the characterization of integral inequalities of such a kind was given in terms of certain lower and upper indices known as Matuszewska-Orlicz indices. Note that in the cited papers the integral inequalities were studied as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M190">View MathML</a>. Such inequalities are also of interest when they allow to impose different conditions as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M190">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M192">View MathML</a>; such a case was dealt with in [22,23].

3 The fractional oscillatory integral operators in the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M193">View MathML</a>

In this section, we are going to use the following statement on the boundedness of the Hardy operator:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M194">View MathML</a>

Theorem G[24]

The inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M195">View MathML</a>

holds for all non-negative and non-increasinggon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M196">View MathML</a>if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M197">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M198">View MathML</a>.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M82">View MathML</a>, andKis a SCZK and the Calderón-Zygmund singular integral operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53">View MathML</a>is of type<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74">View MathML</a>. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75">View MathML</a>and any polynomial<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76">View MathML</a>the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M204">View MathML</a>

holds for any ball<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M205">View MathML</a>and for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M19">View MathML</a>.

Moreover, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79">View MathML</a>andKis a CZK

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M208">View MathML</a>

(3.1)

holds for any ball<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M205">View MathML</a>and for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M210">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M211">View MathML</a>. For arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M212">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M213">View MathML</a> for the ball centered at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M214">View MathML</a> and radius r, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M215">View MathML</a>. We represent f as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M216">View MathML</a>

and have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M217">View MathML</a>

It is known that (see [5], see also [7,25,26]), if K is a SCZK and the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53">View MathML</a> is of type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75">View MathML</a> and any polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76">View MathML</a> the operator S is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M224">View MathML</a> and boundedness of S in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7">View MathML</a> (see [5]) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M226">View MathML</a>

where constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M227">View MathML</a> is independent of f.

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M229">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M230">View MathML</a>. We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M231">View MathML</a>

By Fubini’s theorem and applying Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M232">View MathML</a>

(3.2)

Moreover, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M233">View MathML</a> the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M234">View MathML</a>

(3.3)

is valid. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M235">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M236">View MathML</a>

(3.4)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M237">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79">View MathML</a>. From the weak <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M239">View MathML</a> boundedness of T (see [6]) and (3.4), it follows that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M240">View MathML</a>

(3.5)

Then by (3.4) and (3.5), we get the inequality (3.1). □

Proof of Theorem 1.1

By Lemma 3.1 and Theorem G, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M241">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M211">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M243">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M79">View MathML</a>. □

Proof of Theorem 1.2

The proof of Theorem 1.2 follows from Theorem F and the following observation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M245">View MathML</a>

 □

4 Commutators of fractional oscillatory integral operators in the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M193">View MathML</a>

Let T be a Calderón-Zygmund singular integral operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M100">View MathML</a>. A well known result of Coifman, Rochberg and Weiss [27] states that the commutator operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M248">View MathML</a> is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M59">View MathML</a>. The commutator of Calderón-Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, for example, [2,28,29]).

First, we recall the definition of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M251">View MathML</a>.

Definition 2 Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M210">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M253">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M254">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M255">View MathML</a>

If one regards two functions whose difference is a constant as one, then space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M251">View MathML</a> is a Banach space with respect to norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M257">View MathML</a>.

Remark 1 (1) The John-Nirenberg inequality: there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M258">View MathML</a>, such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M260">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M261">View MathML</a>

(2) The John-Nirenberg inequality implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M262">View MathML</a>

(4.1)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M263">View MathML</a>.

(3) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M264">View MathML</a>. Then there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M227">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M266">View MathML</a>

(4.2)

where C is independent of f, x, r and t.

Lemma 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M100">View MathML</a>, Kis a SCZK and the Calderón-Zygmund singular integral operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53">View MathML</a>is of type<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74">View MathML</a>. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75">View MathML</a>and any polynomial<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76">View MathML</a>the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M273">View MathML</a>

holds for any ball<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M205">View MathML</a>and for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M19">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M211">View MathML</a>. For arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M212">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M213">View MathML</a> for the ball centered at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M214">View MathML</a> and radius r, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M215">View MathML</a>. We represent f as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M281">View MathML</a>

and have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M282">View MathML</a>

It is known that (see [5], see also [7,25,26]), if K is a SCZK and the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M53">View MathML</a> is of type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M74">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M75">View MathML</a> and any polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M76">View MathML</a> the commutator operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M106">View MathML</a> is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M224">View MathML</a> and boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M106">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M7">View MathML</a> (see [5]) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M293">View MathML</a>

where constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M227">View MathML</a> is independent of f.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M295">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M296">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M297">View MathML</a>

Let us estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M298">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M299">View MathML</a>

Applying Hölder’s inequality and by (4.1), (4.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M300">View MathML</a>

In order to estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M301">View MathML</a> note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M302">View MathML</a>

By (4.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M303">View MathML</a>

Thus, by (3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M304">View MathML</a>

Summing up <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M301">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M307">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M308">View MathML</a>

(4.3)

Finally,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M309">View MathML</a>

and statement of Lemma 4.1 follows by (3.4). □

Proof of Theorem 1.3 The statement of Theorem 1.3 follows by Lemma 4.1 and Theorem G in the same manner as in the proof of Theorem G. □

Proof of Theorem 1.4 The proof of Theorem 1.4 follows from the Theorem 7.4 in [11] and the following observation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/70/mathml/M310">View MathML</a>

 □

Competing interests

The author declares that they have no competing interests.

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