Hardy-Type Operators in Lorentz-Type Spaces Defined on Measure Spaces
- PDF / 277,966 Bytes
- 28 Pages / 612 x 792 pts (letter) Page_size
- 7 Downloads / 237 Views
DOI: 10.1007/s13226-020-0453-1
HARDY-TYPE OPERATORS IN LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES1 Qinxiu Sun∗ , Xiao Yu∗∗ and Hongliang Li∗∗∗ ∗ Department
of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, China
∗∗ Department
of Mathematics, Shangrao Normal University, Shangrao 334001, China
∗∗∗ Department
of Mathematics, Zhejiang International Studies University, Hangzhou 310012, China e-mail: [email protected]
(Received 2 June 2018; accepted 12 June 2019) Weight criteria for the boundedness and compactness of generalized Hardy-type operators Z T f (x) = u1 (x) f (y)u2 (y)v0 (y) dµ(y), x ∈ X, (0.1) {φ(y)≤ψ(x)}
in Orlicz-Lorentz spaces defined on measure spaces is investigated where the functions φ, ψ, u1 , u2 , v0 are positive measurable functions. T :
0 ΛG v0 (w0 )
→
1 ΛG v1 (w1 )
and T :
Some sufficient conditions of boundedness of
0 ΛG v0 (w0 )
→ ΛvG11 ,∞ (w1 ) are obtained on Orlicz-Lorentz
spaces. Furthermore, we achieve sufficient and necessary conditions for T to be bounded and compact from a weighted Lorentz space Λpv00 (w0 ) to another Λpv11 ,q1 (w1 ). It is notable that the function spaces concerned here are quasi-Banach spaces instead of Banach spaces. Key words : Hardy operator; Orlicz-Lorentz spaces; weighted Lorentz spaces; boundedness; compactness. 2010 Mathematics Subject Classification : 46E30, 46B42.
1. I NTRODUCTION For the Hardy operator S defined by Sf (x) =
Rx 0
f (t)dt, the weighted Lebesgue-norm inequalities
have been characterized by many authors (e.g. [3, 12, 30, 32]). Sawyer [36] characterized the weights 1
Supported by Natural Science Foundation of Zhejiang Province of China (LY19A010001), National Natural Science
Foundation of China (11961056), Natural Science Foundation of Jiangxi Province of China (20151BAB211002).
1106
QINXIU SUN, XIAO YU AND HONGLIANG LI
u, v such that S : Lp,q (u) → Lr,s (v) is bounded, under certain restriction on exponents p, q, r and s. Later on Carro and Soria [5] described the exponents p0 , p1 , the weights u0 , u1 , w0 , w1 such that S : Λpv00 (w0 ) → Λpu11,∞ (w1 ) or S : Λpu00 (w0 ) → Λpu11 (w1 ) is bounded. Rx For the Hardy operator A by Af (x) = x1 0 f (t)dt, Sawyer [36] analyzed the weights v and w such that A : Lr (v) → Lp,q (w) is bounded under some assumptions on exponents p, q, r. Given non-negative measurable functions ψ and φ on R+ define the operator Z x H1 f (x) = ψ(x) φ(t)f (t)dt, x > 0. 0
Ferreyra [10] gave a characterization of boundedness of H1 : Lr1 (u1 ) → Lp1 ,q1 (w1 ) under the assumptions 1 ≤ r1 ≤ min(p1 , q1 ) and normability of Lp1 ,q1 (w1 ). Edmund, Gurka and Pick in [7, Theorems 3-4] obtained characterization of boundedness and compactness of H1 : Lr0 ,s0 (v0 ) → Lp0 ,q0 (w0 ), when max(r0 , s0 ) ≤ min(p0 , q0 ) and the Lorentz spaces Lr0 ,s0 (v0 ) and Lp0 ,q0 (w0 ) are normable. The result in [7] can also be used to the description of boundedness and compactness of the high dimensional Hardy operator
Z φ(y)f (y)dy, x ∈ Rn ,
Hf (x) = ψ(x)
(1.1)
B(0,|x|)
from Lr,s (u) to L
Data Loading...
