On the non-homogeneous central Morrey type spaces in L¹(Rⁿ) and the weak boundedness of some operators

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  • On the non-homogeneous central Morrey type spaces in $L^{1}(mathbf{R}^{n})$ and the weak boundedness of some operators (The deepening of function spaces and its environment)

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Our aim in this note is to discuss thc weak boundcdncss of thc maximal and generalized Ricsz potential opcrators in thc non-homogeneous central Morrcy type spacc mathbb{J}I^{1q, nu}(R^{gammaiota}) of all measurable functions f on R^{rt} satisfying Vert fVert{lambda f^{1qv}(R^{n})}=(int{1}^{infty}(r^{-nu}Vert fVert {L^{1}(B(0, tau))})^{q}frac{dr}{r})^{1/q}<infty for -infty<v<infty and 0<qleqinfty ; when q=infty, we apply a necessary modification. To do this, wc consider thc family WM^{p, q, nu}(R") of all functions fin L_{foc}^{p}(R^{71}) such that Vert fVert_{Wf_{1}J^{p.q.nu}(R^{??})}=sup_{lambdacdot 0}int_{1} ^{infty}(r^{-nu}lambda|{xin B(0, r):|f(x)|>lambda}|^{{imath}/p})^{q} frac{dr}{r}<infty, where 1leq pleqinfty.

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