A characterization of two weight norm inequalities for fractional and Poisson integrals Academic Article uri icon

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abstract

  • For 1 > p q > 1 > p \leqslant q > \infty and w ( x ) w(x) , v ( x ) v(x) nonnegative functions on R n {{\mathbf {R}}^n} , we show that the weighted inequality \[ ( | T f | q w ) 1 / q C ( f p v ) 1 / p {\left ( {\int {|Tf{|^q}w} } \right )^{1/q}} \leqslant C{\left ( {\int {{f^p}v} } \right )^{1/p}} \] holds for all f 0 f \geqslant 0 if and only if both \[ [ T ( χ Q v 1 p ) ] q w C 1 ( Q v 1 p ) q / p > \int {{{[T({\chi _Q}{v^{1 - p’}})]}^q}w \leqslant {C_1}{{\left ( {\int _Q {{v^{1 - p’}}} } \right )}^{q/p}} > \infty } \] and \[ [ T ( χ Q w ) ] p v 1 p C 2 ( Q w ) p / q > {\int {{{[T({\chi _Q}w)]}^{p’}}{v^{1 - p’}} \leqslant {C_2}\left ( {\int _Q w } \right )} ^{p’/q’}} > \infty \] hold for all dyadic cubes Q Q . Here T T denotes a fractional integral or, more generally, a convolution operator whose kernel K K is a positive lower semicontinuous radial function decreasing in | x | |x| and satisfying K ( x ) C K ( 2 x ) K(x) \leqslant CK(2x) , x R n x \in {{\mathbf {R}}^n} . Applications to degenerate elliptic differential operators are indicated. In addition, a corresponding characterization of those weights v v on R n {{\mathbf {R}}^n} and w w on R + n + 1 {\mathbf {R}}_ + ^{n + 1} for which the Poisson operator is bounded from L p ( v ) {L^p}(v) to L q ( w ) {L^q}(w) is given.

publication date

  • January 1, 1988