Electronic Journal of ifferential Equations, Vol. 207 (207), No. 236,. 8. ISSN: 072-669. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu HEAT AN LAPLACE TYPE EQUATIONS WITH COMPLEX SPATIAL VARIABLES IN WEIGHTE BERGMAN SPACES CIPRIAN G. GAL, SORIN G. GAL Communicated by Jerome A. Goldstein Abstract. In a recent book, the authors of this aer have studied the classical heat and Lalace equations with real time variable and comlex satial variable by the semigrou theory methods, under the hyothesis that the boundary function belongs to the sace of analytic functions in the oen unit disk and continuous in the closed unit disk, endowed with the uniform norm. The urose of the resent note is to show that the semigrou theory methods works for these evolution equations of comlex satial variables, under the hyothesis that the boundary function belongs to the much larger weighted Bergman sace B α() with < +, endowed with a L -norm. Also, the case of several comlex variables is considered. The roofs require some new changes aealing to Jensen s inequality, Fubini s theorem for integrals and the L -integral modulus of continuity. The results obtained can be considered as comlex analogues of those for the classical heat and Lalace equations in L (R) saces.. Introduction Extending the method of semigrous of oerators in solving the evolution equations of real satial variable, a way of comlexifying the satial variable in the classical evolution equations is to comlexify their solution semigrous of oerators, as it was summarized in the book [4]. In the cases of heat and Lalace equations and their higher order corresondents, the results obtained can be summarized as follows. Let = {z C; z < } be the oen unit disk and A() = {f : C; f is analytic on, continuous on }, endowed with the uniform norm f = su{ f(z) : z }. It is well-known that (A(), ) is a Banach sace. Let f A() and consider the oerator W t (f)(z) = + f(ze iu )e u2 /(2t) du, z. (.) In [3] (see also [4, Chater 2], for more details) it was roved that (W t, t 0) is a (C 0 )-contraction semigrou of linear oerators on A() and that the unique 200 Mathematics Subject Classification. 4703, 4706, 4760. Key words and hrases. Comlex satial variable; semigrous of linear oerators; heat equation; Lalace equation; weighted Bergman sace. c 207 Texas State University. Submitted August 25, 207. Published Setember 29, 207.
2 C. G. GAL, S. G. GAL EJE-207/236 solution u(t, z) (that belongs to A(), for each fixed t 0) of the Cauchy roblem is exactly t (t, z) = 2 u 2 ϕ 2 (t, z), (t, z) (0, + ), z = reiϕ, z 0, (.2) In the same contribution [3], setting u(0, z) = f(z), z, f A(), (.3) Q t (f)(z) := t u(t, z) = W t (f)(z). (.4) + f(ze iu ) u 2 du, z, (.5) + t2 we roved that (Q t, t 0) is a (C 0 )-contraction semigrou of linear oerators on A(). Consequently, the unique solution u(t, z) (that belongs to A(), for each fixed t) of the Cauchy roblem is exactly 2 u t 2 (t, z) + 2 u ϕ 2 (t, z) = 0, (t, z) (0, + ), z = reiϕ, z 0, (.6) u(0, z) = f(z), z, f A(), (.7) u(t, z) = Q t (f)(z). (.8) The goal of the resent note is to show that the well-osedness of the above roblems in the sace A(), can be relaced by well-osedness in some (larger) weighted Bergman saces defined in what follows. For 0 < < +, < α < + and ρ α (z) = (α + )( z 2 ) α, the weighted Bergman sace Bα(), is the sace of all analytic functions in, such that [ / f(z) da α (z)] < +, where da α (z) = ρ α (z)da(z), with da(z) = dxdy = rdrdθ, z = x + iy = reiθ, the normalized Lebesgue area measure on the unit disk of the comlex lane. Setting f,α = ( f(z) da α (z)) /, for < +, it is well-known that (Bα(),,α ) is a Banach sace, while for 0 < <, Bα() is a comlete metric sace with metric d(f, g) = f g,α. For other details concerning Bergman saces in the comlex lane and their roerties, we refer the reader to the books [6, Chater, Section.] and [2,. 30-32]. The results obtained can be considered as comlex analogues of those for the classical heat and Lalace equations in L (R) saces (see, e.g., [5,. 23]). In Section 2, we reconsider (.), (.2), (.3), (.4) assuming that the boundary function f Bα() with < +. Section 3 treats (.5), (.6), (.7), (.8) under the same hyothesis for the boundary function f. It is worth mentioning that since the uniform norm used in the case of the sace A() is now relaced with the L -tye norm in the Bergman sace Bα(), the roofs of these results require now some changes based on new tools, like the Jensen s inequality, the Fubini s theorem for integrals and the L -integral modulus of continuity.
EJE-207/236 HEAT AN LAPLACE EQUATIONS IN COMPLEX VARIABLES 3 2. Heat-tye equations with comlex satial variables The first main result of this section is concerned with the heat equation of comlex satial variable. Theorem 2.. Let < + and consider W t (f)(z) given by (.), for z. Then, (W t, t 0) is a (C 0 )-contraction semigrou of linear oerators on B α() and the unique solution u(t, z) that belongs to B α() for each fixed t, of the Cauchy roblem (.2) with the initial condition u(0, z) = f(z), is given by u(t, z) = W t (f)(z). z, f B α(), Proof. By [3, Theorem 2.] (see also [4, Theorem 2.2.,. 27]), W t (f)(z) is analytic in and for all z, t, s 0, we have W t (f)(z) = k=0 a ke k2t/2 z k and W t+s (f)(z) = W t [W s (f)](z). In what follows, we aly the following well-known Jensen tye inequality for integrals: if + G(u)du =, G(u) 0 for all u R and ϕ(t) is a convex function over the range of the measurable function of real variable F, then ( + ) + ϕ F (u)g(u)du ϕ(f (u))g(u)du. Now, since + e u2 /(2t) du =, by the above Jensen s inequality with ϕ(t) = t, F (u) = f(ze iu ) and G(u) = e u2 /(2t) du, we find W t (f)(z) + f(ze iu ) e u2 /(2t) du. Multilying this inequality by ρ α (z) = (α + )( + z 2 ) α, integrating on with resect to da(z) (the normalized Lebesgue s area measure) and taking into account the Fubini s theorem, we obtain W t (f)(z) da α (z) + [ ] f(ze iu ) da α (z) e u2 /(2t) du. But writing z = re iθ in olar coordinates and taking into account that da(z) = rdrdθ, some simle calculations lead to the equality f(ze iu ) da α (z) = f(z) da α (z), for all u R. (2.) This immediately imlies W t (f),α f,α. So W t (f) Bα() and W t is a contraction. Next, for f Bα() let us introduce the integral modulus of continuity (see, e.g., []) ( /. ω (f; δ) B α = su f(ze ih ) f(z) da α (z)) 0 h δ Reasoning as above and taking into account that da(z) = dxdy =, we obtain W t (f)(z) f(z) da α (z)
4 C. G. GAL, S. G. GAL EJE-207/236 Consequently, we get + [ ] f(ze iu ) f(z) da α (z) e u2 /(2t) du + ω (f; u ) /(2t) Bα e u2 du + ω (f; ( t) u e u + ) 2 /(2t) du Bα t t = 2 ω (f; t) B (v + α ) e v2 /2 dv 0 = C ω (f; t) B α. W t (f) f B α C ω (f; t) B α, and so it yields that lim t 0 W t (f) f B α = 0. Since it is well-known that (see, e.g., [6, g. 3, roof of Proosition.2]) the convergence in the Bergman sace means uniform convergence in any comact subset of, it follows that lim t 0 W t (f)(z) = f(z), for all z. Now, let s (0, + ), V s be a small neighborhood of s, both fixed, and take an arbitrary t V s, t s. Alying the reasonings in the roof of [3, Theorem 2., (iii)] (see also [4, Theorem 2.2., (iii)]), we get W t (f)(z) W s (f)(z) + 2 t s f(ze iu ) e u2 /(2t) + /(2s) e u2 du 2s f(ze iu ) e u2 /c 2 2u 2 du, c 2 where c deends on s (and not on t). enoting K s = + /c 2 e u2 2u2 c 4 c du, i.e., where 0 < K 2 s < +, by the roof of [4, Theorem 2.2., (iii)], it follows that W t (f)(z) W s (f)(z) da α (z) ( 2 ) t s K s ( + f(ze iu ) e u2 /c 2 2u 2 K s c 4 c 4 ) daα c 2 du (z) and reasoning exactly as at the beginning of the roof, we obtain W t (f)(z) W s (f)(z) da α (z) ( ) t s Ks f,α. 2 This imlies W t (f) W s (f),α 2 t s K s f,α. Therefore, (W t, t 0) is a (C 0 )-contraction semigrou of linear oerators on Bα(). Also, since the series reresentation for W t (f)(z) is uniformly convergent in any comact disk included in, it can be differentiated term by term with resect to t and ϕ. We then easily obtain that W t (f)(z) satisfies the Cauchy roblem in the statement of the theorem. We also note that in equation we must take z 0 simly because z = 0 has no olar reresentation, namely z = 0 cannot be reresented as function of ϕ. This comletes the roof.
EJE-207/236 HEAT AN LAPLACE EQUATIONS IN COMPLEX VARIABLES 5 The artial differential equation (.2) can equivalently be exressed in terms of t and z as follows. Corollary 2.2. Let < +. For each f Bα(), the initial value roblem t + ( z 2 z + z2 2 u ) z 2 = 0, (t, z) R + \{0}, u(0, z) = f(z), z, is well-osed and its unique solution is W t (f) C (R + ; B α()). Proof. By [3, Theorem 2., (i)], we can comute the generator associated with W t (f), to get ( d dt W t(f)(z) ) t=0 = k=0 k 2 2 a kz k = ( k=0 = z2 2 f (z) z 2 f (z). k(k ) 2 + k 2 )a kz k Therefore, the statement is an immediate consequence of a classical result of Hill (see, e.g., [4, Theorem.2., g. 8]). Remark 2.3. Theorem 2. can be easily extended to functions of several comlex variables, as follows. For 0 < <, let f : n C, f(z, z 2,..., z n ) be analytic with resect to each variable z i, i =,..., n, such that n f(z,..., z n ) da α (z )... da α (z n ) <. We write this by f Bα( n ) and ( f B α ( n ) = f(z,..., z n ) da α (z )... da α (z n ) n becomes a norm on Bα( n ). We call the later the weighted Bergman sace in several comlex variables. Following now the model for the semigrou attached to the real multivariate heat equation (see, e.g., [5, g. 69]), for z, z 2,..., z n, we may define the integral of Gauss-Weierstrass tye by H t (f)(z,..., z n ) = () n/2 f(z e iu,..., z n e iun )e u 2 /(2t) du... du n, n where u = (u,..., u n ) and u = u 2 + + u2 n. One can reason exactly as in the roof of Theorem 2. to deduce that (H t, t 0) is (C 0 )-contraction semigrou of linear oerators on B α( n ) and that u(t, z,..., z n ) = H t (f)(z,..., z n ) is the unique solution of the Cauchy roblem ) / t (t, z,..., z n ) = [ 2 u 2 ϕ 2 (t, z,..., z n ) + + 2 u ϕ 2 (t, z,..., z n ) ], t > 0, n z = r e iϕ,..., z n = r n e iϕn, z,..., z n 0, u(0, z,..., z n ) = f(z,..., z n ). Remark 2.4. Reasoning as in the roof of Corollary 2.2 and calculating ( d dt H t(f)(z,..., z n ), ) t=0
6 C. G. GAL, S. G. GAL EJE-207/236 we easily find that the initial value roblem t + n ( z k + z 2 2 u ) k = 0, (t, z,..., z n ) R + (\{0}) n, 2 z k k= z 2 k u(0, z,..., z n ) = f(z,..., z n ), z,..., z n, is well-osed and its unique solution is H t (f) C (R + ; B α( n )). 3. Lalace-tye equations with comlex satial variables The first main result of this section is concerned with the Lalace equation of a comlex satial variable. Theorem 3.. Let < + and consider Q t (f)(z) given by (.5), for z. Then (Q t, t 0) is a (C 0 )-contraction semigrou of linear oerators on B α() and the unique solution u(t, z) that belongs to B α() for each fixed t, of the Cauchy roblem (.6) with the initial condition is given by u(t, z) = Q t (f)(z). u(0, z) = f(z), z, f B α(), Proof. By [3, Theorem 3.] (see also [4, Theorem 2.3.,. 27]), Q t (f)(z) is analytic in and for all z, t, s 0 we have Q t (f)(z) = k=0 a ke kt z k and Q t+s (f)(z) = Q t [Q s (f)](z). Now, since t + u 2 +t du =, by Jensen s inequality, we get 2 Q t (f)(z) t + f(ze iu ) u 2 + t 2 du, which multilied on both sides by ρ α (z), then integrated on with resect to the Lebesgue s area measure da(z) and alying the Fubini s theorem, gives Q t (f)(z) da α (z) t + [ ] f(ze iu ) da α (z) u 2 + t 2 du. As in the roof of Theorem 3., writing z = re iθ (in olar coordinates) and taking into account that da(z) = rdrdθ, some simle calculations lead to the same equality (2.). Hence, we get Q t (f),α f,α. This imlies that Q t (f) Bα() and that Q t is a contraction. To rove that lim t 0 Q t (f)(z) = f(z), for any f Bα() and z, let f = U + iv, z = re ix be fixed with 0 < r ρ < and denote We can write Q t (f)(z) = t F (v) = U[r cos(v), r sin(v)], G(v) = V [r cos(v), r sin(v)]. + F (x u) t 2 + u 2 du + i t + G(x u) t 2 + u 2 du. From the maximum modulus rincile, when estimating the quantity Q t (f)(z) f(z), for z ρ <, we may take r = ρ. Now, assing to limit as t 0 + and taking into account the roerty in the real case (see, e.g., [5,. 23, Exercise 2.8.8]), we find lim Q t t(f)(z) f(z) lim t 0 t 0 + F (x u) t 2 + u 2 du F (x)
EJE-207/236 HEAT AN LAPLACE EQUATIONS IN COMPLEX VARIABLES 7 t + lim t 0 + G(x u) t 2 + u 2 du G(x) = 0, which holds uniformly with resect to z ρ. Consequently, it follows that lim t 0 Q t (f)(z) = f(z), uniformly in any comact subset of. Now, let s (0, + ), V s be a small neighborhood of s, both fixed, and take an arbitrary t V s, t s. Alying the same reasoning as in the roof of [3, Theorem 3., (ii)] (see also [4, Theorem 2.3., (ii)]), we get Setting i.e., + Q t (f)(z) Q s (f)(z) f(ze iu ) t t 2 + u 2 s s 2 + u 2 du = + t s f(ze iu ) u 2 ts du. (t 2 + u 2 )(s 2 + u 2 ) K s = + u 2 ts (t 2 + u 2 )(s 2 + u 2 ) du, = + u 2 ts du K s (t 2 + u 2 )(s 2 + u 2 ) (where 0 < K s < + ), by the roof of [4, Theorem 2.3., (ii)], it follows that Q t (f)(z) Q s (f)(z) da α (z) ( ) t s Ks ( + f(ze iu ) u 2 ts ) daα du K s (t 2 + u 2 )(s 2 + u 2 (z). ) Alying Jensen s inequality, we obtain Q t (f)(z) Q s (f)(z) da α (z) ( ) t s Ks f,α and so Q t (f) Q s (f),α t s K s f,α. Then (Q t, t 0) is a (C 0 )-contraction semigrou of linear oerators on Bα(). Also, since the series reresentation for Q t (f)(z) is uniformly convergent in any comact disk included in, it can be differentiated term by term, with resect to t and ϕ. We then easily obtain that Q t (f)(z) satisfies the Cauchy roblem in the statement of the theorem. We also note that in equation we must take z 0 simly because z = 0 has no olar reresentation. This comletes the roof. Reasoning exactly as in the roof of [4, Theorem 2.3., (v),. immediately get the following result. 53-54], we Corollary 3.2. Let < +. For each f B α(), the initial value roblem t + z z = 0, (t, z) R + \{0}, u(0, z) = f(z), z, is well-osed and its unique solution is Q t (f) C (R + ; B α()).
8 C. G. GAL, S. G. GAL EJE-207/236 Remark 3.3. The above results can be easily extended to several comlex variables. We may define the comlex Poisson-Cauchy integral by P t (f)(z,..., z n ) = + + Γ((n + )/2) t... f(z (n+)/2 e iu,..., z n e iun ) (t 2 + u 2 + + u2 n) du (n+)/2... du n. Using similar arguments, as in the univariate comlex case, we can rove that the unique solution of the Cauchy roblem 2 u t 2 (t, z,..., z n ) + 2 u ϕ 2 (t, z,..., z n ) + + 2 u ϕ 2 (t, z,..., z n ) = 0, t > 0, n z = r e iϕ,..., z n = r n e iϕn, z,..., z n 0, u(0, z,..., z n ) = f(z,..., z n ), for z,..., z n, f B α( n ) is given by u(t, z,..., z n ) = P t (f)(z,..., z n ). Remark 3.4. Reasoning as in the roof of Corollary 3.2 and calculating ( d dt P t(f)(z,..., z n ), ) t=0 we easily obtain the following result: For each f B α( n ), the initial value roblem n t + ( ) zk = 0, (t, z,..., z n ) R + (\{0}) n, z k k= u(0, z,..., z n ) = f(z,..., z n ), z,..., z n, is well-osed and its unique solution is Q t (f) C (R + ; B α()). References [] O. Blasco; Modulus of continuity with resect to semigrous of analytic functions and alications. J. Math. Anal. Al. 435 (206), 66-626. [2] P. uren, A. Schuster; Bergman Saces. American Mathematical Society, Mathematical Surveys and Monograhs, Rhode Island, 2004. [3] C. G. Gal, S. G. Gal, J. A. Goldstein; Evolution equations with real time variable and comlex satial variables. Comlex Variables Ellitic Equations 53 (2008), 753-774. [4] C. G. Gal, S. G. Gal, J. A. Goldstein; Evolution Equations with a Comlex Satial Variable. World Scientific, New Jersey-London-Singaore-Beijing-Shanghai-Hong Kong-Taiei-Chennai, 204. [5] J. A. Goldstein; Semigrous of Linear Oerators and Alications. Oxford University Press, Oxford, 985. [6] H. Hedenmalm, B. Korenblum, K. Zhu; Theory of Bergman Saces. Sringer-Verlag, New York-Berlin-Heidelberg, 2000. Cirian G. Gal eartment of Mathematics, Florida International University, Miami, FL 3399, USA E-mail address: cgal@fiu.edu Uni- Sorin G. Gal University of Oradea, eartment of Mathematics and Comuter Science, Str. versitatii Nr., 40087 Oradea, Romania E-mail address: galso@uoradea.ro