Old and New Morrey Spaces with Heat Kernel Bounds (original) (raw)
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Boundary Value Problems
Let L =-+ V be a Schrödinger operator on R n , where n ≥ 3 and the nonnegative potential V belongs to the reverse Hölder class RH q 1 for some q 1 > n/2. Let b belong to a new Campanato space θ ν (ρ) and I L β be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b, I L β ] with b ∈ θ ν (ρ) on local generalized Morrey spaces LM α,V,{x 0 } p,ϕ , generalized Morrey spaces M α,V p,ϕ and vanishing generalized Morrey spaces VM α,V p,ϕ associated with Schrödinger operator, respectively. When b belongs to θ ν (ρ) with θ > 0, 0 < ν < 1 and (ϕ 1 , ϕ 2) satisfies some conditions, we show that the commutator operator [b, I L β ] are bounded from LM α,V,{x 0 } p,ϕ 1 to LM α,V,{x 0 } q,ϕ 2 , from M α,V p,ϕ 1 to M α,V q,ϕ 2 and from VM α,V p,ϕ 1 to VM α,V q,ϕ 2 , 1/p-1/q = (β + ν)/n.
The Boundedness of Generalized Fractional Integral Operators on Small Morrey Spaces
Cauchy, 2024
Morrey space was first introduced by C.B. Morrey in 1938 which is the solution space of an elliptic partial differential equation. Morrey space can be said to be a generalization of Lebesgue space. Morrey spaces are generalized into generalized Morrey spaces, small Morrey spaces, weighted Morrey spaces, and Bourgain Morrrey spaces. One of the studies in Morrey space is the boundedness of operators in Morrey space. One such operator is the fractional integral operator. One of the generalizations of fractional integral operators is the generalized fractional integral operator. The generalized fractional integral operator is finite on the generalized Morrrey space. In this study, it is extended on small Morrey spaces. The small Morrey space is the set of locally Lebesgue integrable functions with norm defined supremum over radius of ball ∈ (0,1). This paper aims to prove the boundedness properties of the generality of fractional integral operators on small Morrey spaces using Hedberg-type inequality. The first, in this paper will be discuss to prove Hedberg-type inequality on small Morrey spaces using dyadic decomposition, Hölder inequality, and doubling condition. The proof method used is using Hedberg's inequality. The results imply that the generalized fractional integral operator is finite on small Morrey spaces.
Advances in Difference Equations
Let L =-H n + V be a Schrödinger operator on the Heisenberg group H n , where the nonnegative potential V belongs to the reverse Hölder class RH q 1 for some q 1 ≥ Q/2, and Q is the homogeneous dimension of H n. Let b belong to a new Campanato space θ ν (ρ), and let I L β be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b, I L β ] with b ∈ θ ν (ρ) on central generalized Morrey spaces LM α,V p,ϕ (H n), generalized Morrey spaces M α,V p,ϕ (H n), and vanishing generalized Morrey spaces VM α,V p,ϕ (H n) associated with Schrödinger operator, respectively. When b belongs to θ ν (ρ) with θ > 0, 0 < ν < 1 and (ϕ 1 , ϕ 2) satisfies some conditions, we show that the commutator operator [b, I L β ] is bounded from LM α,V p,ϕ 1 (H n) to LM α,V q,ϕ 2 (H n), from M α,V p,ϕ 1 (H n) to M α,V q,ϕ 2 (H n), and from VM α,V p,ϕ 1 (H n) to VM α,V q,ϕ 2 (H n), 1/p-1/q = (β + ν)/Q.
Boundedness of fractional integral operator with rough kernel on generalized Morrey space
Let Ω be a homogeneous function with zero degree on R n. Fractional integral operator with rough kernel T Ω,α maps function f to convolution between f and kernel function K Ω,α = Ω(x)|x| α−n where 0 < α < n. In this paper, we show Spanne type result and Adam type result for T Ω,α on generalized Morrey spaces. B(z,r) |Ω(z − y)|| f 2 (y)|dy ≤ c sup r>0 B(z,r)∩B c (x,2t) r −n |Ω(z − y)|| f (y)|dy ≤ c sup r>t B(z,r)∩B c (x,2t) |y − z| −n |Ω(z − y)|| f (y)|dy ≤ c sup r>t B(z,r)∩B c (x,2t) |x − y| −n |Ω(z − y)|| f (y)|dy ≤ c B c (x,2t) |Ω(z − y)||x − y| −n | f (y)|dy. By Minkowski inequality, we obtain M Ω f 2 L p (B(x,t)) ≤ c t n/p Ω L p (S n−1) B c (x,2t) |x − y| −n f (y)dy. Note that, inequation (1) also holds for α = 0. Therefore, we have M Ω f 2 L p (B(x,t)) ≤ c t n/p Ω L p (S n−1) ∞ t s −1−(n/p) f L p (B(x,s)) ds.
Old and New Morry Spaces via Heat Kernel Bounds
arXiv (Cornell University), 2006
Given p ∈ [1, ∞) and λ ∈ (0, n), we study Morrey space L p,λ (R n) of all locally integrable complex-valued functions f on R n such that for every open Euclidean ball B ⊂ R n with radius r B there are numbers C = C(f) (depending on f) and c = c(f, B) (relying upon f and B) satisfying r −λ B B |f (x) − c| p dx ≤ C and derive old and new, two essentially different cases arising from either choosing c = f B = |B| −1 B f (y)dy or replacing c by P tB (x) = tB p tB (x, y)f (y)dy-where t B is scaled to r B and p t (•, •) is the kernel of the infinitesimal generator L of an analytic semigroup {e −tL } t≥0 on L 2 (R n). Consequently, we are led to simultaneously characterize the old and new Morrey spaces, but also to show that for a suitable operator L, the new Morrey space is equivalent to the old one.
Morrey spaces and fractional integral operators
Expositiones Mathematicae, 2009
The present paper is devoted to the boundedness of fractional integral operators in Morrey spaces defined on quasimetric measure spaces. In particular, Sobolev, trace and weighted inequalities with power weights for potential operators are established. In the case when measure satisfies the doubling condition the derived conditions are simultaneously necessary and sufficient for appropriate inequalities.
Weak Type Inequalities for Some Integral Operators on Generalized Nonhomogeneous Morrey Spaces
Journal of Function Spaces and Applications, 2013
We prove weak type inequalities for some integral operators, especially generalized fractional integral operators, on generalized Morrey spaces of nonhomogeneous type. The inequality for generalized fractional integral operators is proved by using two different techniques: one uses the Chebyshev inequality and some inequalities involving the modified Hardy-Littlewood maximal operator and the other uses a Hedberg type inequality and weak type inequalities for the modified Hardy-Littlewood maximal operator. Our results generalize the weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces and extend to some singular integral operators. In addition, we also prove the boundedness of generalized fractional integral operators on generalized non-homogeneous Orlicz-Morrey spaces.