scholarly journals A trace inequality for generalized potentials in Lebesgue spaces with variable exponent

2004 ◽  
Vol 2 (1) ◽  
pp. 55-69 ◽  
Author(s):  
David E. Edmunds ◽  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Variable Exponent ◽  
Homogeneous Type ◽  
Trace Inequality ◽  
Spaces Of Homogeneous Type ◽  
Weighted Inequality ◽  
Euclidean Spaces ◽  
Riesz Potentials ◽  
Lebesgue Spaces ◽  
Fractal Sets

A trace inequality for the generalized Riesz potentialsIα(x)is established in spacesLp(x)defined on spaces of homogeneous type. The results are new even in the case of Euclidean spaces. As a corollary a criterion for a two-weighted inequality in classical Lebesgue spaces for potentialsIα(x)defined on fractal sets is derived.

scholarly journals Dyadic Fefferman–Stein inequalities and the equivalence of Haar bases on weighted Lebesgue spaces

2011 ◽  
Vol 141 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Hugo Aimar ◽  
Ana Bernardis ◽  
Luis Nowak
Keyword(s):  
Sufficient Conditions ◽  
General Setting ◽  
Maximal Operators ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Weighted Inequality ◽  
Lebesgue Spaces ◽  
Weighted Lebesgue Spaces ◽  
Dyadic Systems ◽  
Vector Valued

We give sufficient conditions on two dyadic systems to obtain the equivalence of corresponding Haar systems on dyadic weighted Lebesgue spaces on spaces of homogeneous type. In order to obtain these results, we prove a Fefferman–Stein weighted inequality for vector-valued dyadic Hardy–Littlewood maximal operators with dyadic weights in this general setting.

scholarly journals A note on the trace inequality for Riesz potentials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Giorgi Imerlishvili ◽  
Alexander Meskhi
Keyword(s):  
Integrable Function ◽  
Riesz Potential ◽  
Sufficient Condition ◽  
Homogeneous Type ◽  
Trace Inequality ◽  
Spaces Of Homogeneous Type ◽  
Necessary And Sufficient Condition ◽  
Riesz Potentials ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

AbstractWe establish a necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality\lVert I_{\alpha}f\rVert_{L^{p}_{v}(\mathbb{R}^{n})}\leq C\lVert f\rVert_{L^{p% ,1}(\mathbb{R}^{n})}for the Riesz potential {I_{\alpha}}, where {L^{p,1}(\mathbb{R}^{n})} is the Lorentz space. The same problem is studied for potentials defined on spaces of homogeneous type.

scholarly journals Discrete Paraproduct Operators on Variable Hardy Spaces

2019 ◽  
Vol 63 (2) ◽  
pp. 304-317 ◽  
Author(s):  
Jian Tan
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Variable Exponent ◽  
Discrete Version ◽  
Variable Exponents ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Hölder Continuous ◽  
Analysis Techniques ◽  
Calderón’S Reproducing Formula

AbstractLet$p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )$be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of paraproduct operators$\unicode[STIX]{x1D70B}_{b}$on variable Hardy spaces$H^{p(\cdot )}(\mathbb{R}^{n})$, where$b\in \text{BMO}(\mathbb{R}^{n})$. As an application, we show that non-convolution type Calderón–Zygmund operators$T$are bounded on$H^{p(\cdot )}(\mathbb{R}^{n})$if and only if$T^{\ast }1=0$, where$\frac{n}{n+\unicode[STIX]{x1D716}}<\text{ess inf}_{x\in \mathbb{R}^{n}}p\leqslant \text{ess sup}_{x\in \mathbb{R}^{n}}p\leqslant 1$and$\unicode[STIX]{x1D716}$is the regular exponent of kernel of$T$. Our approach relies on the discrete version of Calderón’s reproducing formula, discrete Littlewood–Paley–Stein theory, almost orthogonal estimates, and variable exponents analysis techniques. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

A trace inequality of Riesz potentials in variable exponent Orlicz spaces

2012 ◽  
Vol 285 (11-12) ◽  
pp. 1466-1485
Author(s):  
Yoshihiro Mizuta ◽  
Tetsu Shimomura
Keyword(s):  
Variable Exponent ◽  
Orlicz Spaces ◽  
Trace Inequality ◽  
Riesz Potentials
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