Boundedness of Bessel–Riesz Operator in Variable Lebesgue Measure Spaces

Boundedness of Bessel–Riesz Operator in Variable Lebesgue Measure Spaces

In this manuscript, we establish the boundedness of the Bessel–Riesz operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mrow><mi>α</mi><mo>,...

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Bibliographic Details
Main Authors:	Muhammad Nasir, Ali Raza, Luminiţa-Ioana Cotîrlă, Daniel Breaz
Format:	Article
Language:	English
Published:	MDPI AG 2025-01-01
Series:	Mathematics
Subjects:	variable Lebesgue spaces Bessel–Riesz kernel Bessel–Riesz operator boundedness Hölder’s inequality dyadic decomposition
Online Access:	https://www.mdpi.com/2227-7390/13/3/410
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Summary:	In this manuscript, we establish the boundedness of the Bessel–Riesz operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mrow><mi>α</mi><mo>,</mo><mi>γ</mi></mrow></msub><mi>f</mi></mrow></semantics></math></inline-formula> in variable Lebesgue spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mrow><mi>p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup></semantics></math></inline-formula>. We prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mrow><mi>α</mi><mo>,</mo><mi>γ</mi></mrow></msub><mi>f</mi></mrow></semantics></math></inline-formula> is bounded from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mrow><mi>p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mrow><mi>p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup></semantics></math></inline-formula> and from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mrow><mi>p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mrow><mi>q</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup></semantics></math></inline-formula>. We explore various scenarios for the boundedness of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mrow><mi>α</mi><mo>,</mo><mi>γ</mi></mrow></msub><mi>f</mi></mrow></semantics></math></inline-formula> under general conditions, including constraints on the Hardy–Littlewood maximal operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula>. To prove these results, we employ the boundedness of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula>, along with Hölder’s inequality and classical dyadic decomposition techniques. Our findings unify and generalize previous results in classical Lebesgue spaces. In some cases, the results are new even for constant exponents in Lebesgue spaces.
ISSN:	2227-7390