Integral Operators in Non-Standard Function Spaces: Volume 3: Advances in Grand Function Spaces (Operator Theory: Advances and Applications, 298)

Preface
Contents
18 Integral Operators on Weighted Grand Lebesgue Spaces (WGLS)
	18.1 Boundedness in WGLSs with Non-Doubling Measure
		18.1.1 Hardy–Littlewood Maximal Operator
		18.1.2 Singular Integrals
		18.1.3 Commutators of Singular Integrals
		18.1.4 Unbounded Domains
	18.2 Sobolev-Type Theorem in WGLS on Nonhomogeneous Spaces
		18.2.1 Unbounded Domains
	18.3 Mapping Properties in WGLS of Vector-Valued Functions
		18.3.1 Weighted Grand Bochner–Lebesgue Spaces
			Banach Lattices
			Lattice Hardy–Littlewood Maximal Operator
			Main Results
		18.3.2 Weighted Norm Inequalities for Integral Operators
		18.3.3 More on Singular Integrals
		18.3.4 The Case of Unbounded Space
	18.4 Hardy–Littlewood and Calderón–Zygmund Operators in Abstract Extrapolation Banach Lattices and Their Dual Köthe Spaces
		18.4.1 Calderón–Zygmund Operators in Extrapolation Spaces
	18.5 Applications to Weighted Grand Orlicz–Zygmund Spaces with Ap Weights
	18.6 Comments to Chap.18
19 Integral Operators in Grand Mixed-Normed Function Spaces
	19.1 Weighted Grand Mixed-Normed Lebesgue Spaces (WGMNLSs), Boundedness of Singular Integral Operators
	19.2 Weighted Sobolev-Type Inequalities in WGMNLSs
	19.3 Trace Inequalities to Fractional Integrals on GMNLSs
	19.4 Comments to Chap.19
20 Grand Variable Exponent Function Spaces
	20.1 Preliminaries
		20.1.1 Variable Exponent Lebesgue Spaces (VELSs)
		20.1.2 Grand Variable Exponent Lebesgue Spaces (GVELSs)
		20.1.3 Grand Variable Exponent Morrey Spaces (GVEMSs)
	20.2 Hardy–Littlewood and Singular Integral Operators in Weighted Grand Variable Exponent Lebesgue Spaces
		20.2.1 Auxiliary Statements
			Variable Exponent Muckenhoupt Weights
			Weighted Extrapolation
		20.2.2 Basic Lemmas
		20.2.3 Main Results
		20.2.4 Applications
			Singular Integrals
			Operators on Rectifiable Curves
	20.3 Density, Duality, and Regularity in Lp
		20.3.1 Density of L∞ and Lp(·), λ(·) in Lp(·), λ(·), θ for Finite Measures
		20.3.2 Duality and Predualily in GVELS on σ-Finite Measure Spaces
	20.4 Boundedness of Integral Operators on GVELSs Without the Log-Hölder Continuity Condition on Exponent
		20.4.1 Maximal Operator
		20.4.2 Extrapolation Results
			Extrapolation in Variable Exponent Lebesgue Spaces
			Extrapolation GVELSs
			Extrapolation for Sublinear Operators
		20.4.3 Applications for Operators of Harmonic Analysis
			Sharp Maximal Function
			Calderón–Zygmund Singular Operators
		20.4.4 Applications to Fourier Trigonometric Series and Approximation by Trigonometric Polynomials in Proximinal Subspace of Lp(·),θ
	20.5 Sobolev-Type Inequalities for Potentials in GVELSs
		20.5.1 New Scale of GVELS
		20.5.2 Fractional Integrals: Sobolev-Type Theorems
		20.5.3 Embedding Theorems of Sobolev Type
		20.5.4 Maximal and Calderón–Zygmund Operators
		20.5.5 Commutators
	20.6 One-Sided Integral Operators
		20.6.1 Preliminaries
		20.6.2 One-Sided Extrapolation
		20.6.3 One-Sided Maximal and Calderón–Zygmund Operators
		20.6.4 One-Sided Fractional Integrals
	20.7 Bernstein-Type Inequality in GVELS
		20.7.1 Preliminaries
		20.7.2 The Main Result
	20.8 The Riemann Boundary Value Problem in the Class of Cauchy-Type Integrals with Densities of GVELS
		20.8.1 Statement of the Problem and Some Auxiliary Results
		20.8.2 Main Results: Solution of the Riemann Problem (20.112)
	20.9 Grand Variable Exponent Morrey Spaces (GVEMSs)
		20.9.1 Integral Operators in VEMSs Defined on SHT: Norm Estimates
			Maximal Operator
			Calderón–Zygmund Singular Operator
			Fractional Integral Operators
		20.9.2 Fractional Integrals in VEMS Defined on Nonhomogeneous Space
		20.9.3 Integral Operators in GVEMSs
			Boundedness of Maximal Operators
			Calderón–Zygmund Operators
			Fractional Integrals in GVEMSs Defined on an SHT
			Fractional Integrals in GVEMSs Defined on Nonhomogeneous Space
	20.10 Comments to Chap.20
21 Extrapolation in Grand Function Spaces
	21.1 Preliminaries
		21.1.1 Classical Lorentz Spaces
	21.2 Weighted Extrapolation in GLSs and Applications
		21.2.1 Extrapolation in Weighted GLSs
		21.2.2 Applications in One-Weight Inequalities
	21.3 Extrapolation in (Grand) Banach Function Spaces (BFSs)
		21.3.1 Extrapolation in BFSs
	21.4 Extrapolation in Weighted Classical and Grand Lorentz Spaces with Applications
		21.4.1 Weighted Extrapolation in Lorentz Spaces
		21.4.2 Extrapolation in Grand Lorentz Spaces
		21.4.3 Applications of Extrapolation Results in Grand Lorentz Spaces
			Maximal, Fractional, and Singular Integral Operators
			Commutators
			Further Remarks
		21.4.4 Extrapolation in Grand BFSs
	21.5 Extrapolation Results in Grand Lebesgue Spaces Defined on Product Sets
		21.5.1 Preliminaries
		21.5.2 Weighted Extrapolation: Known Results
		21.5.3 Extrapolation in Weighted Grand Lebesgue Spaces: Main Results
		21.5.4 Some Applications to One-Weight Inequalities
	21.6 Extrapolation in Grand Lebesgue Spaces with Ainfty Weights
		21.6.1 Preliminaries
		21.6.2 The Main Result
		21.6.3 Applications to the Boundedness of Operators of Harmonic Analysis
	21.7 Weighted Extrapolation in Mixed-Norm Spaces and Application
		21.7.1 Preliminaries
		21.7.2 Main Results
			Extrapolation in Mixed-Norm Spaces
		21.7.3 Extrapolation in Some Mixed-Norm Function Spaces
			Mixed-Norm Lebesgue Spaces with Product Weights
		21.7.4 Grand Mixed-Norm Weighted Lebesgue Spaces and Mixed-Norm Grand Weighted Lebesgue Spaces
		21.7.5 The Case of Variable Exponent Spaces
		21.7.6 Some Other Properties of Grand Mixed-Normed Function Spaces
	21.8 Comments to Chap.21
22 Grand Variable Hajłasz–Sobolev and Hölder Spaces
	22.1 Preliminaries
		22.1.1 Grand Variable Exponent Hajłasz–Sobolev and Grand Variable Parameter Hölder Spaces
		22.1.2 Grand Variable Exponent Hajłasz–Morrey Spaces
		22.1.3 Maximal Operator
		22.1.4 Weighted Extrapolation
		22.1.5 Riesz Potentials
		22.1.6 Sharp Maximal Function of Fractional Order
	22.2 Embeddings from GVEHSS to GVPHS
		22.2.1 Embeddings for Spaces Defined on an SHT
		22.2.2 Embeddings in Spaces Defined on Domains in Rn
	22.3 Embeddings from GVEHMS to VPHS
		22.3.1 Regularity of a Fractional Integral Operator in the Framework of GVEMSs
	22.4 Comments to Chap.22
23 Grand Lebesgue-Type Spaces
	23.1 How Much Does the Grand Lebesgue Space Depends on the Choice of Aggrandizer?
		23.1.1 On Coincidence of Grand Spaces with Different Aggrandizers
			On the Embedding Lp)b(Ω)-3mu→Lp)a(Ω)
		23.1.2 Coincidence of Grand Spaces with Aggrandizers of Some Classes
		23.1.3 Scales of Aggrandizers Generating Different Grand Spaces
		23.1.4 On Fourier Multipliers in Grand Spaces
	23.2 Maximal and Fractional Operators in Grand Lebesgue Spaces over Rn
		23.2.1 Grand Spaces Lap),θ(Ω) and Vanishing Grand Space ap),θ(Ω)
		23.2.2 Choice of the Weight a Admitting the Validity of Sobolev Theorem and Boundedness of M in Grand Spaces Lp),θa(Rn) and ap),θ(Rn )
		23.2.3 Denseness of the Classes C0∞ and  in the Vanishing Grand Spaces ap),θ(Rn )
		23.2.4 Denseness of the Class C0∞(Rn)
		23.2.5 Denseness of the Class
		23.2.6 Inversion of the Riesz Fractional Integral in the Vanishing Grand Space ap),θ(Rn )
		23.2.7 Examples of Functions a Satisfying the Condition  aνA∞
	23.3 Weighted Hardy Operators in Grand Lebesgue Spaces over Rn
		23.3.1 On Hardy Operators
			Weighted Estimates
			The Case of Power Weights
		23.3.2 Hardy Operators in Grand Spaces with Integrable Aggrandizers (1+|x|)-λ, λ>n
		23.3.3 The Case of Power Aggrandizers
	23.4 Grand Lebesgue Sequence Spaces
		23.4.1 Grand Space of Sequences
		23.4.2 Operators in Grand Lebesgue Sequence Spaces
		23.4.3 Operators Acting from  Lp),θ1( Ω)  to  q),θ2 ( Zn)  and Vice Versa
		23.4.4 Fractional Calculus in Vanishing Grand Lebesgue Sequence Spaces p),θ(Z)
			Auxiliaries
			Fractional Operators on  Z  with the Semigroup Property
			Discrete Analogue of the Lizorkin Test Function Space
		23.4.5 Weighted Results
			Some Known Weighted Results for Classical p-Spaces
			Boundedness of Weighted Operators in Grand Spaces
	23.5 Grand Lebesgue Space for p=∞ with Application to Riesz Potentials
		23.5.1 Grand L∞)ψ Spaces
		23.5.2 Sobolev-Type Theorem in the Borderline Case αp = n
			Growth of Constant in Hedberg\'s Approach When p→nα
		23.5.3 Main Result for Lebesgue Spaces
		23.5.4 Sobolev–Adams Theorem in the Borderline Case  αp + λ=n for Morrey Spaces
			Growth of Constant in Hedberg\'s Approach When p→n-λα
			Main Result for Morrey Spaces
		23.5.5 Sobolev–Adams Type Theorem in the Borderline Case αp + λ=n for Morrey-Type Spaces
			Growth of Constant in Hedberg\'s Approach When p→n-λα
			Main Result for Morrey-Type Spaces
		23.5.6 Grand L∞)θ Spaces Versus BMO
	23.6 Local Grand Spaces in Quasi-metric Measure Spaces
		23.6.1 Local Grand Lebesgue Spaces Lp),θF,a(Ω,μ).
		23.6.2 Basic Properties
		23.6.3 Power of Distances as Muckenhoupt Weights
		23.6.4 On Operators in Local Grand Lebesgue Spaces
		23.6.5 Maximal Operator
		23.6.6 Singular Operators
		23.6.7 Potential Operators
		23.6.8 An Application to Dirichlet Problem
	23.7 Grand Lebesgue Spaces with Mixed Local and Global Aggrandization
		23.7.1 Grand Spaces with General Aggrandization
			Grand Lebesgue Spaces with Mixed Aggrandization
			On Coincidence of the Mixed Grand Spaces with the Usual Grand Spaces
			Grand Spaces Lp),φa,β(Ω) as New Spaces when (23.105) Is Violated
			Grand Spaces with Specific Aggrandizers
		23.7.2 Boundedness of Operators
			The Maximal Operator
		23.7.3 Singular Integral Operators
	23.8 Commments to Chap.23
References
Symbol Index
	Classes
	Function Spaces
	Miscellaneous
	Operators
Subject Index