FUNCTIONAL ANALYSIS
CONTENTS ABOUT THE GRADIENT OF THE CONFORMAL RADIUS THE IN?UENCE OF THE TUNNEL E?ECT ON ???-TIME DECAY SOME CLASSES OF OPERATORS ON PARTIAL INNER PRODUCT SPACES FROM FORMS TO SEMIGROUPS ACCRETIVE (?)-EXTENSIONS AND REALIZATION PROBLEMS THE DYNAMICAL PROBLEM FOR A NON SELF-ADJOINT HAMILTONIAN EXTENSION OF THE ??-METRIC: THE ???CASE FAMILIES OF HOMOMORPHISMS IN NON-COMMUTATIVE GELFAND THEORY: COMPARISONS AND EXAMPLES OPERATOR SPLITTING WITH SPATIAL-TEMPORAL DISCRETIZATION ON A THEOREM OF KARHUNEN AND RELATED MOMENT PROBLEMS AND QUADRATURE FORMULAE EXPLICIT ERROR ESTIMATES FOR EIGENVALUES OF SOME UNBOUNDED JACOBI MATRICES ALGEBRAIC RE?EXIVITY AND LOCAL LINEAR DEPENDENCE: GENERIC ASPECTS AN INVITATION TO THE ??-FUNCTIONAL CALCULUS NECESSITY OF PARAMETER-ELLIPTICITY FOR MULTI-ORDER SYSTEMS OF DI?ERENTIAL EQUATIONS CANONICAL EIGENVALUE DISTRIBUTION OF MULTILEVEL BLOCK TOEPLITZ SEQUENCES WITH NON-HERMITIAN SYMBOLS CURVATURE INVARIANT AND GENERALIZED CANONICAL OPERATOR MODELS – I ABOUT COMPRESSIBLE VISCOUS FLUID FLOW IN A 2-DIMENSIONAL EXTERIOR DOMAIN ON CANONICAL SOLUTIONS OF A MOMENT PROBLEM FOR RATIONAL MATRIX-VALUED FUNCTIONS MAXIMAL ????-REGULARITY FOR A 2D FLUID-SOLID INTERACTION PROBLEM TRANSFER FUNCTIONS FOR PAIRS OF WANDERING SUBSPACES NON-NEGATIVITY ANALYSIS FOR EXPONENTIAL-POLYNOMIAL- TRIGONOMETRIC FUNCTIONS ON [0, ?) COMPACTNESS OF THE ?-NEUMANN OPERATOR ON WEIGHTED (0, ??)-FORMS GALERKIN METHOD WITH GRADED MESHES FOR WIENER-HOPF OPERATORS WITH PC SYMBOLS IN ????SPACES ON THE SPECTRUM OF SOME CLASS OF JACOBI OPERATORS IN A KREIN SPACE HOLOMORPHY IN MULTICOMPLEX SPACES ON SOME CLASS OF SELF-ADJOINT BOUNDARY VALUE PROBLEMS WITH THE SPECTRAL PARAMETER IN THE EQUATIONS AND THE BOUNDARY CONDITIONS ON THE WELL-POSEDNESS OF EVOLUTIONARY EQUATIONS ON IN?NITE GRAPHS REPARAMETRIZATIONS OF NON TRACE-NORMED HAMILTONIANS = ??, ????? THE QUANTITY ??(??, ??) = ????(??)?= ????(??)?(1 ?????2), ?????, (1 1) IS CALLED THE CONFORMAL RADIUS OF ??AT THE POINT ??= ??(??) WE STUDY THE GRADIENT OF ??DE?NED AS ???(??, ??) = ???(??, ??) ??? + ?????(??, ??) ??? = 2??¯??, ??= ??+ ??????? (1 2) F G AVKHADIEV AND K -J WIRTHS [1], [2] PROVED THAT ???(??, ??) IS A DI?EO- MORPHISM OF ??ONTO A CERTAIN DOMAIN ??(THE TYPE OF ??DEPENDS ON THE TYPE OF DOMAIN ??) IN THE PAPERS [3], [4] WE SHOWED THAT THE GRADIENT OF THE CONFORMAL RADIUS IS A CONFORMAL MAP IF AND ONLY IF ??IS THE UNIT DISK ?? THIS WORK ADDS TO A RECENT PAPER [1] AND EXTENDS RESULTS IN [3], [4] FROM THE POINT OF VIEW OF QUASI-CONFORMAL MAP ????2 ?1 , ?????? AN EQUALITY IS ATTAINED FOR THE FUNCTION ??(??) = ??????1 + ¯???? ??+ ??, ??????, AT EVERY POINT ?????? LEMMA 2 IF THE FUNCTION ??IS REGULAR IN SEMI-PLANE ??= {??: RE ??> 0} AND ??(??) ???, THEN FOR THE DERIVATIVE ??? THE ESTIMATE ????(??)??RE ?? RE ??, ?????, IS VALID THE EQUALITY IS ATTAINED FOR THE FUNCTION ??(??) = ??????+ ?? ?????+ ????, ??, ??, ??, ????, ?????????> 0, AT EVERY POINT ????? 3 MAIN RESULTS LET ????= {??: ?ARG ???< ????/2}, ???(0, 1], AND ??0 = {??: ?IM ???< ??/2} THEOREM 3 1 FOR ANY COMPACT SUBSET OF A CONVEX DOMAIN ??= ??(??), EXCEPT ????, ???[0, 1], THE GRADIENT (1 2) OF THE CONFORMAL RADIUS (1 1) IS A QUASI-CONFORMAL MAPPING FOR THE ANGULAR DOMAIN ????, ???(0, 1], AND THE STRIP ??0 THE IDENTITY ??¯??¯??(??,??) ??¯????(??,??) ?1 IS VALID, AND FOR ANY COMPACT SUBSET OF ????AND ??0 (1 2) IS A DEGENERATE MAP =? ?????1 THUS, THE FUNCTION ??SATIS?ES THE CONDITIONS OF THE EXTENDED SCHWARZ LEMMA LET ?0(??) = 2?? 1??? THEN ??????(??) ???(??) = ?0( ˜??(??)) = ????(??) 1?????(??) AND HENCE ????(??) ???(??) = ??(??) 1 ?????(??) (3 2) WE CALCULATE THE DERIVATIVES ??¯??¯??= { ¯??, ¯??}, ??¯????= ?¯?? 1?????2 IS ATTAINED FOR THE EXTREMAL FUNCTION ??(??) = ????????+?? 1+¯????, ????< 1 IN OUR CASE THIS FUNCTION SATIS?ES THE IDENTITY ???¯??¯?? ??¯??????1 LET US ?ND A FUNCTION ??FOR WHICH THIS IDENTITY IS VALID WE REPLACE THE EXTREMAL FUNCTION ??FROM THE SCHWARZ LEMMA IN (3 2) AND DECOMPOSE THE RIGHT 1 ????(??)?2 ????(??)? ] < 1???2 FUNCTIONAL ANALYSIS EXAMPLE LET DOMAIN ??BE THE SQUARE WITH THE MAPPING FUNCTION ([5], P 77) ??= ??(??) = ??? ???? (1 ???4)1/2 THE CONFORMAL RADIUS FOR THE DOMAIN ??= ??(????) = ????? ?1 ???4??4?1/2 , THEREFORE ??¯??= ??4¯??4 ??? 4? 1 ???4??4 4? (1 ???4 ¯??4)3 , ??¯??¯??= ??4 + ??2 + 1 APPARENTLY, WE WILL HAVE ??2(??(????), ??) < ??1(??(??), ????), 0 < ??< 1, FOR ANY CONVEX DOMAIN ??= ??(??) LET US TURN NOW TO A CLASS ?0 OF FUNCTIONS MAPPING ???= {????> 1} ONTO DOMAINS ???, ?????, WHICH REPRESENT AN EXTERIOR OF SOME CLOSED CONVEX CURVE AND FOR WHICH THE SURFACE OF THE CONFORMAL RADIUS IS CONVEX DOWNWARDS THEN IT HOLDS THAT FUNCTIONAL ANALYSIS THEOREM 3 2 FOR ANY COMPACT SUBSET OF THE DOMAIN ???= ??(???) WITH ?NITE CONVEX COMPLEMENT THE GRADIENT (1 2) OF THE CONFORMAL RADIUS ??(??(???), ??(??)) IS A QUASI-CONFORMAL MAP THERE ARE EXACT ESTIMATORS OF ANALOGS OF QUANTITIES FOR TWO KINDS OF QUASI- CONFORMAL MAPS PROOF THE NECESSARY AND SU?CIENT CONDITION OF THE CONVEXITY OF THE BOUNDARY OF THE DOMAIN ???= ??(???) IS (3 1) ARGUING AS EARLIER, WE GET ??????(??) ???(??) ??0 = ( ??????(??) ???(??) ) = ??(??) : ??? REPEATING THE CALCULATIONS FROM THE PROOF OF THE PREVIOUS THEOREM, WE HAVE ??????(??) ???(??) = ?0(??(??)) = ??(??) ?1, (3 4) AND HENCE ????(??) ???(??) = ??(???1) SINCE THE EXPRESSION FOR THE CONFORMAL RADIUS IS ??(???, ??) = ????(??)?(????2 ?1), ????> 1, ??= ??(??), (3 5) WE ESTIMATE SECOND DERIVATIVES SINCE + ¯?? FURTHER {??, ??} = ? AS ???????????> 1, WE USE LEMMA 1 FOR THE FUNCTION ????, TO DEDUCE THE RELATION ?(??¯??)¯??/(??¯??)????1 WE KNOW (LEMMA 1) THAT THE EQUALITY IS ATTAINED FOR THE EXTREMAL FUNCTION ????(??) = ??????1+¯???? ??+??, WHERE ??(??) IS A FUNCTION WITH A SIMPLE ZERO AT ? BUT IN EX- PRESSION (3 4) FUNCTION 1/??(??) HAS TO HAVE A ZERO OF THE SECOND ORDER THEREFORE < 1 FOR ANY COMPACT SUBSET OF A DOMAIN ??? ??= ??(??? ??), ??? ??= {??: ????> ??> 1}, THE GRADIENT OF THE CONFORMAL RADIUS IS A QUASI-CONFORMAL MAP, AND SUP ????0 ??1(??(??), ??? ??) < 1, 1 < ??< ?, SUP ????0 ??1(??(??), ???) = 1 FOR ??-EQUIPOTENTIAL LINE WE HAVE ??????(????) ???(????) = < 1, WHERE SUP????0 ??2(??(????), ???) = 1 FOR ???1 ? NOW WE CONSIDER DOMAINS ??, ????? SUCH DOMAINS REQUIRE ADDITIONAL ANALYSIS TO THIS END, WE FURTHER SPLIT THEM INTO TWO SEPARATE CLASSES THE ?RST CLASS IS CHARACTERIZED BY THE CONVEXITY OF THE EQUIPOTENTIAL LINES SIMILAR TO THE BOUNDARY CURVE, AND THE OTHER BY NON-CONVEXITY OF SUCH LINES THE ?RST CASE IS, IN FACT, SHOWN IN THEOREM 2 1 IN THE SECOND CASE WE REPLACE THE EQUIPOTENTIAL LINES DE?NED BY THE CIRCLES ????= ??, BY THE EQUIPOTENTIAL LINES DE?NED BY THE CIRCLES ???+ ???= 1 ??? THE FOLLOWING RESULT RE?ECTS THE E?ECT OF SUCH A CHANGE THEOREM 3 3 FOR ANY CLOSED ?NITE DOMAIN FROM ??= ˜??(??), ˜??(1) = ?(????– A CONVEX DOMAIN) THE GRADIENT (1 2) OF THE CONFORMAL RADIUS IS A QUASI-CONFORMAL MAPPING IN PARTICULAR, THIS STATEMENT IS VALID FOR THE CONFORMAL RADIUS ??( ˜??( ˜????), ˜??(??)) FOR A DOMAIN ˜??( ˜????), ˜????= {??: ???+ ????1 ???}, 0 < ??< 1 AS {??: ???+ ????1 ???} ?{??: RE ?????}, ??= 1??? ??, ??= 1??? 1+??, IT IS MORE CONVENIENT TO SPEAK ABOUT A SURFACE OF CONFORMAL RADIUS ??(??(??), ??(??)) = 2 RE ??????(??)?, ??(??) = ˜?? (1 ??? 1 + ?? ) , (3 6) FUNCTIONAL ANALYSIS WHICH IS CONSTRUCTED OVER THE SEMI-PLANE ??= {??: RE ??> 0} (CORRESPONDING MAPPING FUNCTIONS ??(??) FORM A CLASS ?0(??)) PROOF THE NECESSARY AND SU?CIENT CONDITION OF THE CONVEXITY OF A DOMAIN IN THIS CASE HAS THE FORM RE (????(????) ???(????) ) ?0 (3 7) THEREFORE, IT CLEARLY HOLDS ???? ??? = ??, (3 8) WHERE ??(??) IS A FUNCTION SATISFYING THE CONDITIONS OF LEMMA 2 NEXT, WE COMPUTE THE DERIVATIVES OF THE CONFORMAL RADIUS (3 6) WE HAVE ??¯??= ???? ¯??? (? ¯???(??+ ¯??) )? ¯?? = ???? ?¯??? ( ¯???? ¯??? RE ??+ ) , ??¯??¯??= ???? ¯??? ( ?¯??? ( ¯???? ¯??? RE ??+ 2 ? SINCE ??¯????= ????¯??? (? ??? ( ¯???? ¯??? RE ??+ ))? ?? = 2????? ( ???? ??? RE ??+ ???? ??? + ¯???? ¯??? ) , IT HOLDS ????¯????= ???? ??? RE ??+ 2 ? ????2 RE ??+ 2 RE ???RE ??(2?????+ ????2) ????2 RE ??+ 2 RE ??? RE ?? ( 2 RE ?? RE ??+ ????2) ????2 RE ??+ 2 RE ?? = 1 LET US ?ND THE FUNCTION ??FOR WHICH THE EQUALITY ?(??¯??)¯??/(??¯??)????1 IS VALID NOTE THAT THE EQUALITY IN THE INEQUALITY ???? ???2/2???????+ ????2/2, ??2 ?2??> 0 THE SOLUTION OF THE DI?ERENTIAL EQUATION ??? ???2/2 = 2??HAS THE FORM ??= ????+ ?? AMONG ALL SUCH FUNCTIONS WE CHOOSE THE ONE THAT MAPS THE RIGHT HALF-PLANE ON ITSELF, NAMELY ??= 2 ? ????2 RE ??+ 2 RE ?????(??, ??) ???(??, ??) < 1, (3 9) BUT SUP ????0(??)?{??(??,??)} ??(??, ??) = 1 WE CAN CONCRETIZE AN ESTIMATION (3 9), HAVING INCLUDED ??IN ˜??(˜????) FOR SOME ˜???(0, 1) THERE IS NOT A ?NITE EXHAUSTION FOR DOMAIN ??(??), ??(??) ??0(??), BY ?NITE DOMAINS WITH CONVEX BOUNDARIES THEREFORE THERE IS NOT THE ANALOG OF THE COE?CIENT ??2 FROM PROOFS OF PREVIOUS THEOREMS ? REFERENCES [1] F G AVKHADIEV, K -J WIRTHS, THE CONFORMAL RADIUS AS A FUNCTION AND ITS GRADIENT IMAGE ISRAEL J OF MATHEMATICS 145 (2005), 349–374 [2] F G AVKHADIEV, K -J WIRTHS, SCHWARZ-PICK TYPE INEQUALITIES BIRKH¨AUSER VERLAG, 2009 [3] L A AKSENTYEV, A N AKHMETOVA, ON MAPPINGS RELATED TO THE GRADIENT OF THE CON- FORMAL RADIUS IZV VUZOV MATHEMATICS 6 (2009), 60–64 (THE SHORT MESSAGE) [4] L A AKSENTYEV, A N AKHMETOVA, ON MAPPINGS RELATED TO THE GRADIENT OF THE CON- FORMAL RADIUS MATHEMATICAL NOTES 1 (2010), 3–12 [5] G M GOLUZIN, GEOMETRIC THEORY OF THE FUNCTION OF COMPLEX VARIABLES NAUKA, 1966