Weight characterization of the trace inequality for the generalized Riemann-Liouville transform in L^p(x) spaces

Abstract

Abstract. Necessary and sufficient conditions on a weight governing the trace inequality for the Riemann-Liouville transform with variable parameter R α(x) in L p(x) spaces are established provided that p and q satisfy the log-Hölder continuity condition. Weighted criteria for theare also derived.Mathematics subject classification (2010): 46E30, 47B34.

An integro-differential inequality related to the smallest positive eigenvalue of p(x)-Laplacian Dirichlet problem

“…In recent years there has been an increasing interest in the study of various mathematical problems with variable exponent (see for example [1,11,12,14,17]). Differential equations and variational problems with p(x)-growth conditions arise from the study of elastic mechanics, oscillation problem, electrorheological fluids or image restoration ( [5,6,16]).…”

Riemann-Liouville and Higher Dimensional Hardy Operators for NonNegative Decreasing Function inLp(·)Spaces

“…Furthermore Hardy type inequality was studied in [12,13] by Rafeiro and Samko in Lebesgue spaces with variable exponent. Weighted problems for the Riemann-Liouville transform in ( ) spaces were explored in the papers [5,[14][15][16] (see also the monograph [17]).…”

Riemann-Liouville and higher dimensional Harday operators for non-negative decreasing function in $L^{p(\cdot)}$ spaces

“…Weighted problems for the Riemann-Liouville transform in L p(x) spaces were explored in the papers [10], [11], [2], [14] (see also the monograph [18]).…”