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Necessary and sufficient conditions on a weight governing the trace inequality for the Riemann-Liouville transform with variable parameter Rα(x) in L p(x) spaces are established provided that p and q satisfy the log-Holder continuity condition. Weighted criteria for the compactness of Rα(x) from Lp(x) to L q(x) v are also derived. Mathematics subject classification (2010): 46E30, 47B34.

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