Some weighted norm inequalities concerning the schrödinger operators (original) (raw)
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SUNTO. Si presentono alcuni risultati sugli operatori massimali associati alle misure superficiali in R n . E. M. Stein [9] ha iniziato lo studio di quest'argomento, dimostrando una disuguaglianza a priori per la funzione massimale sferica, mediante l'utilizzo delle funzioni "g". In [3], M. Cowling e G. Mauceri hanno generalizzato il risultato di Stein. Qui si dà una dimostrazione diversa e forse più semplice della stima a priori, utilizzando la transformata di Mellin, come suggerito in un altro lavoro di Cowling e Mauceri [2].
Proceedings of the American Mathematical Society, 1993
Suppose Φ \Phi is an appropriate Young’s function and w ( x ) , v ( x ) w(x),v(x) are nonnegative locally integrable functions. Let T T denote one of three linear operators of special importance that map suitable functions on R n {R^n} into functions on R n {R^n} . For the Hardy operator T T , we study the inequality \[ ∫ 0 ∞ Φ ( | T f ( x ) | ) w ( x ) d x ⩽ C ∫ 0 ∞ Φ ( | f ( x ) | ) v ( x ) d x \int _0^\infty {\Phi (|Tf(x)|)w(x)\,dx \leqslant C\int _0^\infty {\Phi (|f(x)|)v(x)\,dx} } \] and for the Hardy-Littlewood maximal operator or fractional integrals T T , we discuss the inequalities \[ ∫ R n Φ ( | T ( f v ) ( x ) | ) w ( x ) d x ⩽ C ∫ R n Φ ( | f ( x ) | ) v ( x ) d x . \int _{{R^n}} {\Phi (|T(fv)(x)|)w(x)\,dx \leqslant C\int _{{R^n}} {\Phi (|f(x)|)v(x)\,dx.} } \] In all cases we obtain the necessary and sufficient conditions.
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Proceedings of the American Mathematical Society, 1993
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