Stefan Samko (born 28 March 1941, Rostov-on-Don; Russia) is one of the top leading mathematicians in the world in the field of fractional calculus and in nonstandard function spaces, being the co-author of the encyclopedic monographs in these areas, cf. [29, 30, 75]. For a detailed account on his research, up to his 60th anniversary, see [27], where 149 scientific papers and 3 monographs are mentioned. For a detailed summary on S. Samko’s role in the development of the theory of fractional differential equations and related integral operators, up to his 70th anniversary, consult [58], see also [14, 28] for other articles related with his scientific achievements and biographical information.

Stefan Samko defended, under the supervision of the Academician Fyodor Dmitrievich Gakhov, his Ph.D. thesis in the Belarusian State University in 1967 and his Dr.Sc. degree in 1978 in the Steklov Mathematical Institute. S. Samko began working at the Department of Differential Equations in Rostov State University (later changed to Differential and Integral Equations Department), established in 1961 by Professor F. D. Gakhov. At this department, he passed all the steps from Assistant to Professor and Head of the department. During his work, Stefan Grigorievich developed and taught a number of general education and special courses, including “Mathematical analysis,” “Differential equations,” “Equations of mathematical physics,” “Generalized functions,” “Spherical harmonics,” “Interpolation of linear operators,” and “Differential and integral operators of fractional order.”

Notwithstanding his age, Stefan Samko is still very active in research. His research output encompasses over 300 scientific papers and 5 research monographs and their translations. In what follows, we briefly describe his results obtained in the last decade.

Classical Morrey spaces and their several variants. Classical Morrey-type spaces have been playing an important role in the research of Stefan Samko in the last decade. Embeddings, approximation results, and the study of classical operators of harmonic analysis on such spaces are examples of topics studied by Stefan Samko, see [1,2,3,4,5].

His research topics in this field include the embedding results between local generalized Morrey spaces and weighted Lebesgue spaces, and also between generalized Morrey spaces and generalized Stummel classes, jointly obtained with A. Almeida in [5]. Together with the same author, in [3, 4], Stefan Samko has established approximation results in Morrey spaces, where, in particular, the closure of nice functions in Morrey norm was described in terms of new explicit vanishing properties. This research led to the introduction of new vanishing Morrey subspaces, and in [1, 2] S. Samko established various results on the boundedness, on those subspaces, of a wide class of classical operators, such as maximal, potential, singular, and Hardy-type operators.

In [6,7,8], jointly with F. Deringoz and V.S. Guliyev, S. Samko studied generalized Orlicz–Morrey spaces (gOMs) and weak gMOs. They obtained the boundedness of several operators, viz. the fractional maximal operator, the maximal operator, and their corresponding commutator with BMO-coefficients in vanishing gOMs. The case of Calderón–Zygmund operators is also studied. It should be pointed out that conditions for the boundedness of the operators are given in terms of the so-called supremal operators involving the Young functions. A. Karapetyants and S. Samko [26] introduced and studied mixed norm Bergman–Orlicz–Morrey spaces and proved the boundedness of the Bergman projection and gave a description of functions in these spaces via the behavior of their Taylor coefficients. Similar results, by the same authors, were obtained [23] for the case of mixed norm Bergman–Morrey-type spaces on the unit disc. The general theory of such spaces was constructed in [18] by the same authors.

In [35], D. Lukkassen, L.-E. Persson, and S. Samko studied the weighted \(p\rightarrow q\)-boundedness of the multidimensional weighted Hardy-type operator in the generalized complementary Morrey spaces (gcMs). It was also shown that gcMs are embedded between weighted Lebesgue spaces. S. Samko obtained in [62] that the generalized local Morrey spaces are embedded between weighted Lebesgue spaces with weights differing only by a logarithmic factor, which shows that the generalized global Morrey spaces are embedded between two generalized Stummel classes whose characteristics similarly differ by a logarithmic factor.

In [50], jointly with H. Rafeiro, S. Samko studied several types of embeddings between local Morrey type spaces into Lebesgue spaces. The embeddings for global Morrey-type spaces are also provided into Stummel spaces. Reverse embeddings are also obtained.

Herz spaces and generalizations. In the paper [68], S. Samko introduced the so-called continual variable exponent Herz spaces where all the three characteristics of the space are variable. In the same paper, he proved that sublinear operators, satisfying some size condition, are bounded in the aforementioned spaces. This includes the cases of maximal and Calderón–Zygmund singular operators. In [53], together with H. Rafeiro, he found the conditions on the variable characteristics of the variable exponent Herz space to insure the boundedness of the Riesz potential operator. The result is obtained both in the homogeneous and inhomogeneous spaces.

In [47], H. Rafeiro and S. Samko defined local and global generalized Herz spaces. They established that Morrey-type spaces and complementary Morrey-type spaces are included into the scale of these Herz spaces. The boundedness of a class of sublinear operators in generalized Herz spaces with application to Morrey-type spaces and their complementary spaces, based on the mentioned inclusion, was also shown. Similar results, where the generalized local and global Herz spaces have all their characteristics variable, are studied in [45].

Classes of operators in complex analysis and holomorphic Hölder spaces. In recent works by A. Karapetyants and S. Samko, some classes of operators in complex analysis have been studied, which, among other operators, contain classical operators of mathematical physics. For example, in [21], the mentioned authors together with K. Zhu introduced and studied a class of Hausdorff–Berezin operators on the unit disc. This class includes the Berezin transform itself, as well as some other classical operators, such as the invariant Green’s potential. The authors obtain boundedness conditions for such operators, and discuss questions of approximations of functions by constructions in the form of Hausdorff–Berezin operators.

Furthermore, A. Karapetyants and S. Samko in [17] introduced a convolution form, in terms of integration over the unit disc \(\mathbb {D},\) for operators on holomorphic functions which correspond to Taylor expansion multipliers. They demonstrate the advantages of the introduced integral representation in the study of mapping properties of such operators. In particular, they prove the Young theorem for Bergman spaces in terms of integrability of the kernel of the convolution. A special attention is paid to explicit integral representation of fractional integration and differentiation. Another important advantage is the study of mapping properties of a class of such operators in Hölder-type spaces of holomorphic functions, which in fact is hardly possible when the operator is defined just in terms of multipliers. The authors show, among other results, that for a class of fractional integral operators such a mapping between Hölder spaces is onto.

In greater generality, these results are obtained in the article [20] of the same authors, which considers Hadamard–Bergman operators of variable order and generalized Hölder classes constructed via modulus of continuity of variable order.

The study of operators in Hölder spaces undoubtedly precedes the study of the spaces themselves, which is done in the work [16], in which such spaces are studied in the framework of general domains in the complex plane.

Holomorphic Bergman-type spaces and operators in these spaces. The spaces of holomorphic functions of the Bergman–Orlicz, Bergman–Morrey type, and various other modifications and generalizations on the unit disc and half plane of the complex plane are studied in the papers [15, 22] by A. Karapetyants, H. Rafeiro, and S. Samko (see also the paper [19] by A. Karapetyants and S. Samko). The boundedness of the Bergman projector, the behavior of functions from these spaces near the boundary, and other questions are investigated. The main idea of these papers is the transfer of results and methods for spaces of nonstandard growth known in real analysis to a complex context.

Variable exponent spaces. S. Samko and V. Kokilashvili [31] obtained the boundedness of the Cauchy singular integral operator in weighted variable exponent spaces on Lyapunov curves or curves of bounded rotation without cusps and variable exponent Muckenhoupt weights.

A. Karapetyants and S. Samko in [24] introduced mixed norm variable exponent Bergman spaces on the unit disc, where they use the \(\ell ^q\) norm of the sequence whose entries are variable exponent \(L^{p(\cdot )}(I)\)-norms of the Fourier coefficients of the function f. They proved, among other results, the boundedness of the Bergman projection in the newly introduced space.

In [51], H. Rafeiro and S. Samko, using some pointwise estimates, obtained boundedness results for operators with rough kernels, such as the maximal operator, the fractional maximal operator, the sharp maximal operators, and the fractional operators in the variable exponent Lebesgue and Morrey spaces. The same approach was applied, in [48], to generalized variable exponent Morrey spaces, generalized Orlicz–Morrey, complementary generalized Morrey, or variable exponent Herz spaces to obtain the boundedness of the maximal operator with rough kernel.

In the paper [49], jointly with H. Rafeiro, S. Samko obtained the conditions for the boundedness of a class of sublinear operators on the variable exponent Morrey-type spaces (veMts). A priori assumptions on this class are that the operators are bounded in variable exponent Lebesgue spaces (veLs) and satisfy some size condition. This class includes, in particular, the maximal operator, singular operators with the standard kernel, and the Hardy operators. An embedding of weighted veMts into weighted veLs was also obtained.

S. Samko studied, together with V. S. Rabinovich in [41], the singular integral operator acting on the weighted variable exponent Lebesgue spaces on certain Carleson curves. They also obtained necessary and sufficient conditions for Fredholmness and index formula. Using the techniques for pseudodifferential and Mellin-type operators, the same authors in [40], obtained precise criteria for Fredholmness.

In [54], H. Rafeiro and S. Samko introduced variable exponent Campanato spaces (veCs) in the framework of spaces of homogeneous type and studied embedding results between veCs and variable exponent Morrey spaces or variable exponent Hölder spaces. In [52], they studied the boundedness of the Riesz potential of variable order over bounded domains in the Euclidean space from variable exponent Morrey spaces to veCs. A special attention was devoted to weaken assumptions on variability of the Riesz potential.

S. Samko obtained several other results in the framework of variable exponent spaces. In [65], he studied the mapping properties of Mellin convolution operators in variable exponent Lebesgue spaces. In [66], S. Samko obtained the boundedness of the fractional operator of variable order on a bounded open set in a quasimetric measure space in the borderline case, from variable exponent Lebesgue spaces into BMO. In the paper [67], he studied Hardy inequalities with power weights in the framework of variable exponent setting and, using the aforementioned inequalities, he derived the corresponding Carleman–Knopp inequalities. Estimates of weighted norm of potential kernels truncated to balls, where the admitted class of weights are of radial type, are studied in [69] and the conditions on the validity of the estimates are characterized either in terms of Zygmund-type inequalities on the weights or in terms of their Matuszewska–Orlicz indices. In [64], he obtained a certain modular variable exponent Hardy-type inequality in the n-dimensional Euclidean space with a precise constant via the divergence theorem. He also studied, in [60], the \(p\rightarrow q\) boundedness in variable exponent Lebesgue spaces of the fractional operator of variable order on a bounded open set in a quasimetric measure space with the measure satisfying the growth condition.

S. Samko studied the mapping properties of the Riesz fractional integral operator of variable order on a bounded set in the borderline case from variable exponent Lebesgue spaces into BMO spaces in [59]. Similar results, on the case of variable exponent Morrey spaces, were obtained by H. Rafeiro and S. Samko in [44]. In the same paper, for the Riesz potential with constant order, they also obtained boundedness from variable exponent vanishing Morrey spaces (vevMs) into VMO, the result being new even for the space with constant characteristics of the vevMs. The boundedness of the Riesz potential of variable order from vevMs to VMO was obtained in [43].

S. Samko, jointly with D. Lukkassen and L.-E. Persson in [36], studied weighted multidimensional Hardy-type operators with variable order acting from a variable exponent locally generalized Morrey space (velgMs) to another velgMs. The conditions in terms of Zygmund-type integral inequalities were given for the validity of the aforementioned boundedness.

V.S. Guliyev and S.G. Samko investigated in [12], the boundedness of the Hardy–Littlewood and singular integral operators in generalized variable exponent Morrey spaces (gveMs). Spanne and Adams-type theorems for fractional maximal operators and potential type operators of variable order for gveMs were also obtained. The boundedness conditions were formulated either in terms of Zygmund type integral inequalities or in terms of supremal operators. The same authors, in [11], partially extended these results in the framework of quasimetric measure space instead of the Euclidean space. The previous authors, together with J.J. Hasanov, obtained in [10] similar results in the case of local complementary variable exponent Morrey-type spaces. In [13], M.G. Hajibayov and S. Samko examined the boundedness of generalized Riesz potential operators acting from the variable exponent Lebesgue space into a certain Musielak–Orlicz space in the framework of quasimetric measure space with doubling measure.

In [79], S. Samko and S. Umarkhadzhiev studied a multidimensional integral equation of the first kind with potential-type kernel of small order \(M^{\alpha }\). The integral equation \(M^{\alpha } \varphi (x)= f(x)\) was reduced to an equation of the second kind via the regularization \(R {M}^\alpha \varphi = \varphi + A \varphi\), where A is a compact operator in the space of summable functions of variable order and R is an hypersingular operator.

Grand type spaces. In [70, 77, 78], S. Samko and S. Umarkhadzhiev introduced the notion of grand Lebesgue spaces on sets with infinite measure via aggrandizers. The authors also studied the boundedness of Riesz potential in the newly introduced space in [71]. The notion of grand Morrey-type spaces, both mixed and partial versions, is introduced in [76] and the boundedness of some operators is obtained.

A. Karapetyants and S. Samko [25] defined the notion of grand and small Bergman spaces of holomorphic functions in the unit disc and the boundedness of the Bergman projection on grand Bergman spaces was proved.

In [32], S. Samko and V. Kokilashvili obtained necessary and sufficient conditions for the boundedness of the weighted singular integral operator with power weights in grand Lebesgue spaces. Fredholmness for a singular integral operator in Carleson curves was also obtained.

The concept of grand Lebesgue sequence spaces was introduced by H. Rafeiro, S. Umarkhadzhiev, and S. Samko in [55]. The boundedness of several operators of harmonic analysis—maximal, convolution, Hardy, Hilbert, and fractional operators—was obtained. Special attention was paid to fractional calculus, including the density of the discrete version of a Lizorkin sequence test space in vanishing grand spaces.

In [56], the previous authors defined the grand Lebesgue space corresponding to the case \(p= \infty\) and similar grand spaces for Morrey and Morrey type spaces, also for \(p=\infty\). The newly introduced spaces were used to study mapping properties for the Riesz potential operator in the borderline cases.

In [57], H. Rafeiro, S. Samko, and S. Umarkhadzhiev, following the paper [73] by the last two named authors, introduced local grand Lebesgue spaces, over a quasimetric measure space, where the Lebesgue space is “aggrandized” not everywhere but only at a given closed null set F. They obtained the boundedness of several operators—maximal operator, singular operators with standard kernel, potential type operator—and gave an application to Dirichlet problem for the Poisson equation, taking F as the boundary of the domain.

Assorted results. Others results obtained by Stefan Samko include, jointly with E. Liflyand in [33], a Leray-type formula on the Fourier transform of a radial function; a general result for integral operators with homogeneous kernels and corresponding sharp constants which allow to obtain a plethora of inequalities like some new multidimensional Hardy–Hilbert-type inequalities with D. Lukkassen and L.-E. Persson in [37, 38]; several resolving formulas of a hypersingular integral equation to a problem of cracks in elasticity theory in [63]; and introduction of a Chen-type modification of Hadamard fractional integro-differentiation and application to \(L^p\) convergence with M.U. Yakhshiboev in [72]. He also studied, in [9] jointly with L. Diening and [74], mapping properties of Riesz potential in generalized Hölder spaces with a given dominant of continuity modulus. S. Samko and L.-E. Persson in [39] obtained new sharp constants in Hardy-type inequalities both in one- and multidimensional cases.

It is also noteworthy to mention that, in the last decade, S. Samko published a handful of survey papers, viz., on the topic of Wiener algebra of absolutely convergent Fourier integrals with E. Liflyand and R.Trigub in [34]; on the issue of fractional operations of integration and differentiation of variable order, both in the Euclidean setting and in the framework of quasimetric measure spaces in [61]; on mapping properties of fractional integrals of variable order and function spaces with variable exponents with H. Rafeiro in [46]; and on results regarding Morrey–Campanato spaces with respect to the properties of the spaces themselves jointly with H. Rafeiro and N. Samko in [42].

Stefan Samko has been actively participating in editorial and organizational work. For many years, he has been a member of the editorial boards of several mathematical journals, e.g., Izvestiya vuzov. Mathematics, Vladikavkaz Mathematical Journal, Integral Transformations and Special Functions, Fractional Calculus and Applied Analysis, Transactions of A. Razmadze Mathematical Institute. He has regularly been the chairman and member of organizing committees of a number of international conferences and symposiums. It is worth pointing out that several international conferences were held in his honor.

Stefan Samko continues his intensive mathematical activity, showing an example of optimism, responsibility, and passion in science, inspiring his students and colleagues on new searches and discoveries.

We wish Stefan Samko good health and continued success in his research!

Alexandre Almeida, Zalina A. Kusraeva, Humberto Rafeiro