Assume d > 3 and that 2 is an open, bounded and convex set with volume...

  

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Assume d > 3 and that 2 is an open, bounded and convex set with volume 2| of Rd. Denote diam(2) = sup{|x - yl : x, y e 2}. a) Show that setting Tf(x) = [fu)dy |x - yld-1 defines bounded linear operator from L?(2) to itself. You may find useful to apply Young's convolution inequality IIf * g|l, < IIF1I,l|g||, V(f, g) € L°(N) × L°(N), for all p, q, r > 1 with ! 1-1. + %3D Assume d > 3 and that 2 is an open, bounded and convex set with volume 2| of Rd. Denote diam(2) = sup{|x - yl : x, y e 2}. a) Show that setting Tf(x) = [fu)dy |x - yld-1 defines bounded linear operator from L?(2) to itself. You may find useful to apply Young's convolution inequality IIf * g|l, < IIF1I,l|g||, V(f, g) € L°(N) × L°(N), for all p, q, r > 1 with ! 1-1. + %3D

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