Yuri I. Karlovich

Carleson curves, Muckenhoupt weights, and Toeplitz operators

1 Carleson curves.- 1.1 Definitions and examples.- 1.2 Growth of the argument.- 1.3 Seifullayev bounds.- 1.4 Submultiplicative functions.- 1.5 The W transform.- 1.6 Spirality indices.- 1.7 Notes and comments.- 2 Muckenhoupt weights.- 2.1 Definitions.- 2.2 Power weights.- 2.3 The logarithm of a Muckenhoupt weight.- 2.4 Symmetric and periodic reproduction.- 2.5 Portions versus arcs.- 2.6 The maximal operator.- 2.7 The reverse Holder inequality.- 2.8 Stability of Muckenhoupt weights.- 2.9 Muckenhoupt condition and W transform.- 2.10 Oscillating weights.- 2.11 Notes and comments.- 3 Interaction between curve and weight.- 3.1 Moduli of complex powers.- 3.2 U and V transforms.- 3.3 Muckenhoupt condition and U transform.- 3.4 Indicator set and U transform.- 3.5 Indicator functions.- 3.6 Indices of powerlikeness.- 3.7 Shape of the indicator functions.- 3.8 Indicator functions of prescribed shape.- 3.9 Notes and comments.- 4 Boundedness of the Cauchy singular integral.- 4.1 The Cauchy singular integral.- 4.2 Necessary conditions for boundedness.- 4.3 Special curves and weights.- 4.4 Brief survey of results on general curves and weights.- 4.5 Composing curves and weights.- 4.6 Notes and comments.- 5 Weighted norm inequalities.- 5.1 Again the maximal operator.- 5.2 The Calderon-Zygmund decomposition.- 5.3 Cotlars inequality.- 5.4 Good ? inequalities.- 5.5 Modified maximal operators.- 5.6 The maximal singular integral operator.- 5.7 Lipschitz curves.- 5.8 Measures in the plane.- 5.9 Cotlars inequality in the plane.- 5.10 Maximal singular integrals in the plane.- 5.11 Approximation by Lipschitz curves.- 5.12 Completing the puzzle.- 5.13 Notes and comments.- 6 General properties of Toeplitz operators.- 6.1 Smirnov classes.- 6.2 Weighted Hardy spaces.- 6.3 Fredholm operators.- 6.4 Toeplitz operators.- 6.5 Adjoints.- 6.6 Two basic theorems.- 6.7 Hankel operators.- 6.8 Continuous symbols.- 6.9 Classical Toeplitz matrices.- 6.10 Separation of discontinuities.- 6.11 Localization.- 6.12 Wiener-Hopf factorization.- 6.13 Notes and comments.- 7 Piecewise continuous symbols.- 7.1 Local representatives.- 7.2 Fredholm criterion.- 7.3 Leaves and essential spectrum.- 7.4 Metamorphosis of leaves.- 7.5 Logarithmic leaves.- 7.6 General leaves.- 7.7 Index and spectrum.- 7.8 Semi-Fredholmness.- 7.9 Notes and comments.- 8 Banach algebras.- 8.1 General theorems.- 8.2 Operators of local type.- 8.3 Algebras generated by idempotents.- 8.4 An N projections theorem.- 8.5 Algebras associated with Jordan curves.- 8.6 Notes and comments.- 9 Composed curves.- 9.1 Extending Carleson stars.- 9.2 Extending Muckenhoupt weights.- 9.3 Operators on flowers.- 9.4 Local algebras.- 9.5 Symbol calculus.- 9.6 Essential spectrum of the Cauchy singular integral.- 9.7 Notes and comments.- 10 Further results.- 10.1 Matrix case.- 10.2 Index formulas.- 10.3 Kernel and cokernel dimensions.- 10.4 Spectrum of the Cauchy singular integral.- 10.5 Orlicz spaces.- 10.6 Mellin techniques.- 10.7 Wiener-Hopf integral operators.- 10.8 Zero-order pseudodifferential operators.- 10.9 Conformal welding and Hasemans problem.- 10.10 Notes and comments.

Convolution operators and factorization of almost periodic matrix functions

1 Convolution Operators and Their Symbols.- 2 Introduction to Scalar Wiener-Hopf Operators.- 3 Scalar Wiener-Hopf Operators with SAP Symbols.- 4 Some Phenomena Caused by SAP Symbols.- 5 Introduction to Matrix Wiener-Hopf Operators.- 6 Factorization of Matrix Functions.- 7 Bohr Compactification.- 8 Existence and Uniqueness ofAPFactorization.- 9 Matrix Wiener-Hopf Operators withAPWSymbols.- 10 Matrix Wiener-Hopf Operators withSAPWSymbols.- 11 Left Versus Right Wiener-Hopf Factorization.- 12 Corona Theorems.- 13 The Portuguese Transformation.- 14 Some Concrete Factorizations.- 15 Scalar Trinomials.- 16 Toeplitz Operators.- 17 Zero-Order Pseudodifferential Operators.- 18 Toeplitz Operators with SAP Symbols on Hardy Spaces.- 19 Wiener-Hopf Operators with SAP Symbols on Lebesgue Spaces.- 20 Hankel Operators on Besicovitch Spaces.- 21 Generalized AP Factorization.- 22 Canonical Wiener-Hopf Factorization via Corona Problems.- 23 Canonical APW Factorization via Corona Problems.

Weighted norm inequalities

In this chapter we prove that if Γ is a Carleson curve and w is a weight in A P (Γ) (1 <p < ∞), then the Cauchy singular integral operator S is bounded on L P (Γ,w). There are now various proofs of this deep result, and the proof given in the following is certainly not the most elegant proof. However, it is reasonably self-contained and it contains several details which are usually disposed of as “standard” and are therefore omitted in the advanced texts on this topic.

(Semi)-Fredholmness of Convolution Operators on the Spaces of Bessel Potentials

The consideration of above mentioned operators on the union of intervals and/or rays is reduced to the canonical situation of operators W k on L P (ℝ+) with semi almost periodic presymbols K at the expense of inflating the size of K. The Fredholm theory (that is, conditions of n-, d-normality and the index formula) is developed. In particular, relations between (semi-)Fredholmness of W K , invertibility of \({W_{{k_ \pm }}}\) with K ± being almost periodic representatives of K at ±∞, and factorability of K ± are established.

Toeplitz and singular integral operators on Carleson curves with logarithmic whirl points

We consider Toeplitz operators with piecewise continuous symbols and singular integral operators with piecewise continuous coefficients onLp (Γ,w) where 1<p<∞,w is a Muckenhoupt weight and Γ belongs to a large class of Carleson curves. This class includes curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. Our main result says that the essential spectrum of a Toeplitz operator is obtained from the essential range of its symbol by joining the endpoints of each jump by a certain spiralic horn, which may degenerate to a usual horn, a logarithmic spiral, a circular arc or a line segment if the curve Γ and the weightw behave sufficiently well at the point where the symbol has a jump. This result implies a symbol calculus for the closed algebra of singular integral operators with piecewise continuous coefficients onLp (Γ,w).

Poly-Bergman Projections and Orthogonal Decompositions of L2-spaces Over Bounded Domains

The paper is devoted to obtaining explicit representations of poly-Bergman and anti-poly-Bergman projections in terms of the singular integral operators S D and S D * on the unit disk D, studying relations between different true poly-Bergman and true anti-poly-Bergman spaces on the unit disk that are realized by the operators S D and S D * , establishing two new orthogonal decompositions of the space L 2(U, dA) (in terms of poly-Bergman and anti-poly-Bergman spaces) for an arbitrary bounded open set U ⊂ ℂ with the Lebesgue area measure dA, considering violation of Dzhuraev’s formulas and establishing explicit forms of the Bergman and anti-Bergman projections for several open sectors.

Convolution Type Operators with Symbols Generated by Slowly Oscillating and Piecewise Continuous Matrix Functions

The paper deals with a local study of the Banach algebra A [SO, PC] generated by the convolution type operators W a b = a F −1bF with data a E ∈ [SO,PC] n× n and b∈ [SO p ,PC] p n× n which act on the Lebesgue space L p n (ℝ) (1 < p < ∞, n ≥ 1). Here [SO,PC] n× n means the C*-algebra generated by slowly oscillating (SO)and piecewise continuous(PC) n × n matrix functions, and SOBA11020200787 is a Fourier multiplier analogue onL p (ℝ)of [SO,PC] n× n The work is based on the study of Fourier multiplier analogue SO p ofSOon the characterization of the multiplicative linear functionals of slowly oscillating functions, and on the compactness of the commutators AW a, b — W a, b A, where A ∈ A PC] a ∈ SOandb ∈ SO p. Making use of the Allan-Douglas local principle we construct homomorphisms of A [SO PC]onto local Banach algebras and establish a Fredholm criterion for operators A ∈ A [SO PC] in terms of the invertibility of their images in the local algebras.

Nonlocal Singular Integral Operators with Slowly Oscillating Data

The paper is devoted to studying the Fredholmness of (nonlocal) singular integral operators with shifts N = (Σ a g + V g )P + + (Σa g − V g )P − on weighted Lebesgue spaces L p (Γ,w) where 1 < p < ∞, Γ is an unbounded slowly oscillating Carleson curve, w is a slowly oscillating Muckenhoupt weight, the operators P ± = 1/2 (I ± S Γ) are related to the Cauchy singular integral operator SΓ, a g ± are slowly oscillating coefficients, V g are shift operators given by V g f = f o g, and g are slowly oscillating shifts in a finite subset of a subexponential group G acting topologically freely on Γ. The Fredholm criterion for N consists of two parts: of an invertibility criterion for polynomial functional operators A ± = Σa g ± V g in terms of invertibility of corresponding discrete operators on the space l p (G), and of a condition of local Fredholmness of N at the endpoints of Γ established by applying Mellin pseudodifferential operators with compound slowly oscillating V (ℝ)-valued symbols where V (ℝ) is the Banach algebra of absolutely continuous functions of bounded total variation on ℝ.

ONE-SIDED INVERTIBILITY OF BINOMIAL FUNCTIONAL OPERATORS WITH A SHIFT ON REARRANGEMENT-INVARIANT SPACES

AbstractLet Γ be an oriented Jordan smooth curve and α a diffeomorphism of Γ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertibility of the binomial functional operator