Norms of Maximal Functions Between Generalized and Classical Lorentz Spaces
[ X ]
Tarih
2023
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Inst Math & Mechanics Azerbaijan
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
The aim of the paper is to find the norm of the generalized maximal operator M.,.a(b), defined for all measurable functions f on R-n, with 0 < alpha < infinity and functions b, phi : (0, infinity) -> (0, infinity), by M(phi, Lambda alpha(b))f(x) := sup (Q(sic)x)
f chi Q
Lambda(alpha)(b)/phi(|Q|), x is an element of R-n, from generalized Lorentz spaces G Gamma(p, m, v) into classical Lorentz spaces Lambda(q)(w). In order to achieve the goal, we reduce the problem to the solution of the inequality (integral(infinity)(0) [T(u,b)f *(y)](q) w(y) dy)(1/q) <= C (integral(infinity)(0) (integral(x)(0) [f *(s)]p ds) (m/p) v(x) dx)(1/m) where w and v are weight functions on (0,infinity). Here f* is the non-increasing rearrangement of a measurable function f defined on R-n and T-u,T- b is the iterated Hardy-type operator involving suprema, which is defined for a measurable non-negative function g on (0,infinity) by (T(u,b)g)(t) := sup(tau is an element of[t,infinity)) u(tau)/B(tau) integral(tau)(0) g(s)b(s) ds, t is an element of(0,infinity), where u and b are weight functions on (0,infinity) such that u is continuous on (0,infinity) and the function B(t) := integral(t)(0) b(s) ds satisfies 0 < B(t) < infinity for every t is an element of (0,infinity).
f chi Q
Lambda(alpha)(b)/phi(|Q|), x is an element of R-n, from generalized Lorentz spaces G Gamma(p, m, v) into classical Lorentz spaces Lambda(q)(w). In order to achieve the goal, we reduce the problem to the solution of the inequality (integral(infinity)(0) [T(u,b)f *(y)](q) w(y) dy)(1/q) <= C (integral(infinity)(0) (integral(x)(0) [f *(s)]p ds) (m/p) v(x) dx)(1/m) where w and v are weight functions on (0,infinity). Here f* is the non-increasing rearrangement of a measurable function f defined on R-n and T-u,T- b is the iterated Hardy-type operator involving suprema, which is defined for a measurable non-negative function g on (0,infinity) by (T(u,b)g)(t) := sup(tau is an element of[t,infinity)) u(tau)/B(tau) integral(tau)(0) g(s)b(s) ds, t is an element of(0,infinity), where u and b are weight functions on (0,infinity) such that u is continuous on (0,infinity) and the function B(t) := integral(t)(0) b(s) ds satisfies 0 < B(t) < infinity for every t is an element of (0,infinity).
Açıklama
Anahtar Kelimeler
generalized maximal functions, classical and generalized Lorentz spaces, iterated Hardy inequalities involving suprema, weights
Kaynak
Azerbaijan Journal of Mathematics
WoS Q Değeri
Q3
Scopus Q Değeri
Q2
Cilt
13
Sayı
2
