## Publications

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1996
Karlovich, Alexei Yu. "Algebras of singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces." Mathematische Nachrichten. 179 (1996): 187-222. AbstractWebsite

We consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces $$L_M(\Gamma)$$, which are generalizations of the Lebesgue spaces $$L_p(\Gamma)$$, $$1 < p < \infty$$. We suppose that $$\Gamma$$ belongs to a large class of Carleson curves, including curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. For the singular integral operator associated with the Riemann boundary value problem with a piecewise continuous coefficient $$G$$, we establish a Fredholm criterion and an index formula in terms of the essential range of $$G$$ complemented by spiralic horns depending on the Boyd indices of $$L_M(\Gamma)$$ and contour properties. Our main result is a symbol calculus for the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients on $$L_M^n(\Gamma)$$.

Карлович, Алексей Об алгебре сингулярных интегральных операторов в рефлексивных пространствах Орлича на кривых Карлесона. Краевые задачи, специальные функции и дробное исчисление. Минск: Издательство Университетское, 1996.02_1996_gahov-90_minsk-96.pdf
1997
Karlovich, Alexei Yu. "Singular integral operators with regulated coefficients in reflexive Orlicz spaces." Siberian Mathematical Journal. 38.2 (1997): 253-266.Website
1998
Karlovich, Alexei Yu. "Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces." Integral Equations and Operator Theory. 32 (1998): 436-481. AbstractWebsite

The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some disintegration condition{''} which combines properties of spaces and curves, the Boyd and spirality indices. We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic leaves) between the endpoints of jumps. These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra $$\mathfrak{A}$$ generated by SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIO's from $$\mathfrak{A}$$ in terms of their symbols.

Karlovich, Alexei Yu. "The index of singular integral operators in reflexive Orlicz spaces." Mathematical Notes. 64.3 (1998): 330-341. AbstractWebsite

We consider the Banach algebra $$\mathfrak{A}$$ of singular integral operators with matrix piecewise continuous coefficients in the reflexive Orlicz space $$L_M^n(\Gamma)$$. We assume that $$\Gamma$$ belongs to a certain wide subclass of the class of Carleson curves; this subclass includes curves with cusps, as well as curves of the logarithmic spiral type. We obtain an index formula for an arbitrary operator from the algebra $$\mathfrak{A}$$ in terms of the symbol of this operator.

2000
Karlovich, Alexei Yu. "On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces." Integral Equations and Operator Theory. 38 (2000): 28-50. AbstractWebsite

In this paper we extend necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces [K98] to the weighted case $$X(\Gamma,w)$$. These conditions are formulated in terms of indices $$\alpha(Q_tw)$$ and $$\beta(Q_tw)$$ of a submultiplicative function $$Q_tw$$, which is associated with local properties of the space, of the curve, and of the weight at the point $$t\Gamma$$. Using these results we obtain a lower estimate for the essential norm $$S$$ of the Cauchy singular integral operator $$S$$ in reflexive weighted rearrangement-invariant spaces $$X(\Gamma, w)$$ over arbitrary Carleson curves $$\Gamma$$:
$|S|\ge\cot(\pi\lambda_{\Gamma,w}/2)$
where $$\lambda_{\Gamma,w} :=inf_{t\in\Gamma} min\{\alpha(Q_tw), 1 - \beta(Q_tw)\}$$. In some cases we give formulas for computation of $$\alpha(Q_tw)$$ and $$\beta(Q_tw)$$.

2001
Karlovich, Alexei Yu. "Criteria for one-sided invertibility of a functional operator in rearrangement-invariant spaces of fundamental type." Mathematische Nachrichten. 229 (2001): 91-118. AbstractWebsite

Let $$\gamma$$ be a simple open smooth curve and $$\alpha$$ be an orientation-preserving diffeomorphism of $$\gamma$$ onto itself which has only two fixed points. Criteria for one-sided invertibility of the functional operator
$A=aI-bW,$
where $$a$$ and $$b$$ are continuous functions, $$I$$ is the identity operator, $$W$$ is the shift operator: $$(Wf)(t)=f[\alpha(t)]$$, in a reflexive rearrangement-invariant space of fundamental type $$X(\gamma)$$ with nontrivial Boyd indices, are obtained.

Karlovich, Alexei Yu., and Lech Maligranda. "On the interpolation constant for Orlicz spaces." Proceedings of the American Mathematical Society. 129 (2001): 2727-2739. AbstractWebsite

In this paper we deal with the interpolation from Lebesgue spaces $$L^p$$ and $$L^q$$, into an Orlicz space $$L^\varphi$$, where $$1 \le p < q \le \infty$$ and $$\varphi^{-1}(t)=t^{1/p}\rho(t^{1/q-1/p})$$ for some concave function $$\rho$$, with the special attention to the interpolation constant $$C$$. For a bounded linear operator $$T$$ in $$L^p$$ and $$L^q$$, we prove modular inequalities, which allow us to get the estimate, for both, the Orlicz norm and the Luxemburg norm,
$\|T\|_{L^\varphi\to L^\varphi} \le C\max\Big\{ \|T\|_{L^p\to L^p}, \|T\|_{L^q\to L^q} \Big\},$
where the interpolation constant $$C$$ depends only on $$p$$ and $$q$$. We give estimates for $$C$$, which imply $$C<4$$. Moreover, if either $$1 < p < q \le 2$$ or $$2 \le p < q < \infty$$, then $$C< 2$$. If $$q=\infty$$, then $$C\le 2^{1-1/p}$$, and, in particular, for the case $$p=1$$ this gives the classical Orlicz interpolation theorem with the constant $$C=1$$.

2002
Karlovich, Alexei Yu. "Algebras of singular integral operators on rearrangement-invariant spaces and Nikolski ideals." The New York Journal of Mathematics. 8 (2002): 215-234. AbstractWebsite

We construct a presymbol for the Banach algebra $$\operatorname{alg}(\Omega, S)$$ generated by the Cauchy singular integral operator $$S$$ and the operators of multiplication by functions in a Banach subalgebra $$\Omega$$ of $$L^\infty$$. This presymbol is a homomorphism $$\operatorname{alg}(\Omega,S)\to\Omega\oplus\Omega$$ whose kernel coincides with the commutator ideal of $$\operatorname{alg}(\Omega,S)$$. In terms of the presymbol, necessary conditions for Fredholmness of an operator in $$\operatorname{alg}(\Omega,S)$$ are proved. All operators are considered on reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.

Karlovich, Alexei Yu., and Yuri I. Karlovich. "Invertibility in Banach algebras of functional operators with non-Carleman shifts." Ukrains'kyj matematychnyj kongres -- 2001. Pratsi. Sektsiya 11. Funktsional'nyj analiz. Kyiv: Instytut Matematyky NAN Ukrainy, 2002. 107-124. Abstract13_2002_ukrainian_math_congress-kyiv-01.pdf

We prove the inverse closedness of the Banach algebra $$\mathfrak{A}_p$$ of functional operators with non-Carleman shifts, which have only two fixed points, in the Banach algebra of all bounded linear operators on $$L^p$$. We suppose that $$1 \le p \le \infty$$ and the generators of the algebra $$\mathfrak{A}_p$$ have essentially bounded data. An invertibility criterion for functional operators in $$\mathfrak{A}_p$$ is obtained in terms of
the invertibility of a family of discrete operators on $$l^p$$. An effective invertibility criterion is established for binomial difference operators with $$l^\infty$$ coefficients on the spaces $$l^p$$. Using the reduction to binomial difference operators, we give effective criteria of invertibility for binomial functional operators on the spaces $$L^p$$.

Karlovich, Alexei Yu., and Yuri I. Karlovich. "One-sided invertibility of binomial functional operators with a shift on rearrangement-invariant spaces." Integral Equations and Operator Theory. 42 (2002): 201-228. AbstractWebsite

Let $$\Gamma$$ be an oriented Jordan smooth curve and $$\alpha$$ a diffeomorphism of $\Gamma$ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertibility of the binomial functional operator $$A=aI-bW$$ where $a$ and $b$ are continuous functions, $$I$$ is the identity operator, $$W$$ is the shift operator, $$Wf=f\circ\alpha$$, on a reflexive rearrangement-invariant space $$X(\Gamma)$$ with Boyd indices $$\alpha_X,\beta_X$$ and Zippin indices $$p_X,q_X$$ satisfying inequalities
$0<\alpha_X=p_X\le q_X=\beta_X<1.$

Karlovich, Alexei Yu. "Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights." Journal of Operator Theory. 47 (2002): 303-323. AbstractWebsite

In this paper we extend results on Fredholmness of singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces $$X(\Gamma)$$ with nontrivial Boyd indices $$\alpha_X,\beta_X$$ [K98] to the weighted case. Suppose a weight $$w$$ belongs to the Muckenhoupt classes $$A_{\frac{1}{\alpha_X}}(\Gamma)$$ and $$A_{\frac{1}{\beta_X}}(\Gamma)$$. We prove that these conditions guarantee the boundedness of the Cauchy singular integral operator $$S$$ in the weighted rearrangement-invariant space $$X(\Gamma,w)$$. Under a disintegration condition'' we construct a symbol calculus for the Banach algebra generated by singular integral operators with matrix-valued piecewise continuous coefficients and get a formula for the index of an arbitrary operator from this algebra. We give nontrivial examples of spaces, for which this disintegration condition'' is satisfied. One of such spaces is a Lebesgue space with a general Muckenhoupt weight over an arbitrary Carleson curve.

2003
Karlovich, Alexei Yu., and Yuri I. Karlovich. "Compactness of commutators arising in the Fredholm theory of singular integral operators with shifts." Factorization, Singular Operators and Related Problems. Eds. Stefan Samko, Amarino Lebre, and António Ferreira dos Santos. Dordrecht: Kluwer Academic Publishers, 2003. 111-129. Abstract

The paper is devoted to the compactness of the commutators $$aS_\Gamma - S_\Gamma aI$$ and $$W_\alpha S_\Gamma - S_\Gamma W_\alpha$$, where $$S_\Gamma$$ is the Cauchy singular integral operator, $$a$$ is a bounded measurable function, $$W_\alpha$$ is the shift operator given by $$W_\alpha f = f\circ\alpha$$, and $$\alpha$$ is a bi-Lipschitz homeomorphism (shift). The cases of the unit circle and the unit interval are considered. We prove that these commutators are compact on rearrangement-invariant spaces with nontrivial Boyd indices if and only if the function a or, respectively, the derivative of the shift a has vanishing mean oscillation.

Karlovich, Alexei Yu. "Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces." Journal of Integral Equations and Applications. 15.3 (2003): 263-320. AbstractWebsite

We prove necessary conditions for the Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces $$L^{p(\cdot)}_w$$ (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever $$w$$ is a Khvedelidze weight, and the variable exponent $$p(t)$$ satisfies the estimate $$|p(\tau)-p(t)|\le A/(-\log|\tau-t|)$$.

Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "Invertibility of functional operators with slowly oscillating non-Carleman shifts." Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, 142. Eds. Albrecht Böttcher, Marinus A. Kaashoek, Amarino Brites Lebre, António Ferreira dos Santos, and Frank-Olme Speck. Basel: Birkhäuser, 2003. 147-174. Abstract

We prove criteria for the invertibility of the binomial functional operator
$A=aI-bW_\alpha$
in the Lebesgue spaces $$L^p(0,1)$$, $$1 < p < \infty$$, where $$a$$ and $$b$$ are continuous functions on $$(0,1)$$, $$I$$ is the identity operator, $$W_\alpha$$ is the shift operator, $$W_\alpha f=f\circ\alpha$$, generated by a non-Carleman shift $$\alpha:[0,1]\to[0,1]$$ which has only two fixed points $$0$$ and $$1$$. We suppose that $$\log\alpha'$$ is bounded and continuous on $$(0,1)$$ and that $$a,b,\alpha'$$ slowly oscillate at $$0$$ and $$1$$. The main difficulty connected with slow oscillation is overcome by using the method of limit operators.

2004
Karlovich, Alexei Yu. "Singular integral operators with flip and unbounded coefficients on rearrangement-invariant spaces." Functional Analysis and its Applications. Proceedings of the international conference, dedicated to the 110th anniversary of Stefan Banach, Lviv National University, Lviv, Ukraine, May 28--31, 2002. Eds. V. Kadets, and W. Zelazko. Amsterdam: Elsevier, 2004. 123-131. Abstract

We prove Fredholm criteria for singular integral operators of the form
$P_++M_bP_-+M_uUP_-,$
where $$P_\pm$$ are the Riesz projections, $$U$$ is the flip operator, and $$M_b,M_u$$ are operators of multiplication by functions $$b,u$$, respectively, on a reflexive rearrangement-invariant space with nontrivial Boyd indices over the unit circle. We assume a priori that $$M_b$$ is bounded, but $$M_u$$ may be unbounded. The function $$u$$ belongs to a class of, in general, unbounded functions that relates to the Douglas algebra $$H^\infty+C$$.

Karlovich, Alexei Yu. "Some algebras of functions with Fourier coefficients in weighted Orlicz sequence spaces." Operator Theoretical Methods and Applications to Mathematical Physics. The Erhard Meister Memorial Volume. Operator Theory: Advances and Applications, 147. Eds. Israel Gohberg, Wolfgang Wendland, António Ferreira dos Santos, Frank-Ollme Speck, and Francisco Sepúlveda Teixeira. Basel: Birkhäuser, 2004. 287-296. Abstract

In this paper, the author proves that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to $$\ell^1\cap\ell^\Phi_{\varphi,w}$$ (resp. to $$\ell^1\cap\ell^\Psi_{\psi,\varrho}$$), where $$\ell^\Phi_{\varphi,w}$$ and $$\ell^\Psi_{\psi,\varrho}$$ are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences
$\varphi=\{\varphi_n\},\quad \psi=\{\psi_n\},\quad w=\{w_n\},\quad \varrho=\{\varrho_n\}$
increase and satisfy the $$\Delta_2$$-condition.

Karlovich, Alexei Yu. "Norms of Toeplitz and Hankel operators on Hardy type subspaces of rearrangement-invariant spaces." Integral Equations and Operator Theory. 49 (2004): 43-64. AbstractWebsite

We prove analogues of the Brown-Halmos and Nehari theorems on the norms of Toeplitz and Hankel operators, respectively, acting on subspaces of Hardy type of reflexive rearrangement-invariant spaces with nontrivial Boyd indices.

2005
Karlovich, Alexei Yu. "Algebras of singular integral operators on Nakano spaces with Khvedelidze weights over Carleson curves with logarithmic whirl points." Izvestiya Vysshih Uchebnyh Zavedeniy. Severo-Kavkazskiy Region. Estestvennye Nauki. Special Issue "Pseudodifferential equations and some problems of mathematical physics". Rostov-on-Don: Rostov University Press, 2005. 135-142. Abstract22_2005_simonenko-70.pdf

We establish a Fredholm criterion for an arbitrary operator in the Banach algebra of singular integral operators
with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over Carleson curves with logarithmic whirl points.

Karlovich, Alexei Yu., and Andrei K. Lerner. "Commutators of singular integrals on generalized Lp spaces with variable exponent." Publicacions Matematiques. 49.1 (2005): 111-125. AbstractWebsite

A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized Lp spaces with variable exponent.

Karlovich, Alexei Yu., and Pedro A. Santos. "On asymptoties of Toeplitz determinants with symbols of nonstandard smoothness." Journal of Fourier Aanalysis and Applications. 11.1 (2005): 43-72. AbstractWebsite

We prove Szegö's strong limit theorem for Toeplitz determinants with a symbol having a nonstandard smoothness. We assume that the symbol belongs to the Wiener algebra and, moreover, the sequences of Fourier coefficients of the symbol with negative and nonnegative indices belong to weighted Orlicz classes generated by complementary $$N$$-functions both satisfying the $$\Delta_2^0$$-condition and by weight sequences satisfying some regularity, and compatibility conditions.

2006
Karlovich, AY. "Higher order asymptotics of Toeplitz determinants with symbols in weighted Wiener algebras." Journal of Mathematical Analysis and Applications. 320.2 (2006): 944-963. AbstractWebsite

We extend a result of Bottcher and Silbermann on higher order asymptotics of determinants of block Toeplitz matrices with symbols in Wiener algebras with power weights to the case of Wiener algebras with general weights satisfying natural submultiplicativity, monotonicity, and regularity conditions.

2007
Karlovich, Alexei Yu. "Algebras of singular integral operators with piecewise continuous coefficients on weighted Nakano spaces." The Extended Field of Operator Theory. Operator Theory: Advances and Applications, 171. Ed. Michael A. Dritschel. Basel: Birkhäuser, 2007. 171-188. Abstract

We find Fredholm criteria and a formula for the index of an arbitrary operator in the Banach algebra of singular integral operators with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over either Lyapunov curves or Radon curves without cusps. These results localize'' the Gohberg-Krupnik Fredhohn theory with respect to the variable exponent.

Karlovich, Alexei Yu. "Asymptotics of block Toeplitz determinants generated by factorable matrix functions with equal partial indices." Mathematische Nachrichten. 280 (2007): 1118-1127. AbstractWebsite

We prove asymptotic formulas for block Toeplitz matrices with symbols admitting right and left Wiener-Hopf factorizations such that all partial indices are equal to some integer number. We consider symbols and Wiener-Hopf factorizations in Wiener algebras with weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. Our results complement known formulas for Holder continuous symbols due to Bottcher and Silbermann.