An L p di erentiable non-di erentiable function

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1 An L di erentiable non-di erentiable function J. Marsall As Abstract. Tere is a a set E of ositive Lebesgue measure and a function nowere di erentiable on E wic is di erentible in te L sense for every ositive at eac oint of E. For every 2 (0; ] and every ositive integer k tere is a set E = E(k; ) of ositive measure and a function wic for every q < as k L q Peano derivatives at every oint of E desite not aving an L kt derivative at any oint of E. A real-valued function f of a real variable is di erentiable at x if tere is a real number f 0 (x) suc tat jf (x + ) f (x) f 0 (x) j = o () as! 0: Fix 2 (0; ). A function is di erentiable in te L sense at x if tere is a real number f 0 (x) suc tat f (x + ) f (x) f 0 (x) = o () as! 0; R =. were kg ()k = jg (t)j dt We ave an in nite family of generalized rst derivatives indexed by te arameter : Most generalized derivatives are not equivalent to te ordinary derivative at a single oint, but many are equivalent on an almost everywere basis. For examle, te symmetric derivative, de ned by fs 0 f(x+) f(x ) (x) = lim!0 2, is zero for te absolute value function at x = 0 even toug tat function is not di erentiable at x = 0, but tis enomenon wic occurs at te single oint x = 0 never occurs on a set of ositive measure: tere cannot exist a set of ositive measure E and a function g so tat gs 0 (x) exists at all oints of E and g 0 (x) exists at no oints of E.[K, age 27] In tis sense te symmetric derivative is equivalent to ordinary di erentiation. So a natural question to ask ere is weter in tis sense te various L derivatives are di erent from ordinary di erentiation and from one anoter. Te oint of tis aer is to answer yes to tis question. If < 2 and f is L 2 di erentiable at x, ten f is L di erentiable at x; since by Holder s inequality, f (x + ) f (x) f 0 2 (x) 2 2 f (x + ) f (x) f 0 2 (x) 2 = o () 2000 Matematics Subject Classi cation. Primary 26A27; secondary 26A24. Key words and rases. L derivative, Peano derivative, L Peano derivative, suer density. Tis researc was artially suorted by a grant from te Faculty and Develoment Program of te College of Liberal Arts and Sciences, DePaul University. Tis aer is in nal form and no version of it will be submitted for ublication elsewere.

2 2 J. MARSHALL ASH so tat f 0 (x) exists and equals f 0 2 (x). It may be useful to tink of a scale of derivatives indexed by, te iger te value of, te better te beavior. Te best beavior, ordinary di erentiability, occurs wen =. Sometimes te scale is extended by lacing te aroximate derivative at = 0. A function f as a kt Peano derivative at x if tere are real numbers f i (x) ; i = 0; ; 2; : : : ; k; suc tat f (x + ) f 0 (x) f (x) f k (x) k k! = o k as! 0: Fix 2 (0; ). A function f as a kt Peano derivative in te L sense at x if tere are real numbers f i (x) ; i = 0; ; 2; : : : ; k; suc tat f (x + ) f 0 (x) f (x) f k (x) k k! = o k as! 0: Te same -scale mentioned for rst derivatives also olds for kt Peano ones as well. Watever te value of k, wen 6= q, L kt order Peano di erentiability is not a.e. equivalent to L q kt order Peano di erentiability; tis is te content of Teorems 2 and 3 below. Te rst extensive discussion of te L Peano derivative tat I am aware of aeared in reference [CZ]. Di erentiation in te L sense for te caracteristic function of a set is very closely related to te concet of suer density, wic is discussed in reference [LMZ]. Teorem. Tere is a set E of ositive Lebesgue measure and a function nowere di erentiable on E wic is di erentiable in te L sense for every ositive at eac oint of E. Proof. Note tat te caracteristic function of te rational numbers rovides a trivial examle since it is nowere di erentiable, but is L di erentiable to 0 at every irrational oint. To avoid suc a triviality, we furter secify tat every element of te equivalence class de ning te L function sould also fail to be di erentiable on E, i.e. canging te function on a set of measure 0 sould not imrove te di erentiability of te function. Order te rational numbers into a sequence and for n = ; 2; : : :, let G n be an oen interval centered at te nt rational of lengt 2 n2. Let C be te comlement of [ i G i. Since j[ i G i j P 2 n2 <, jcj =. Let be te caracteristic function of C. Let I (x; ) = [x ; x + ].. is not di erentiable at almost every oint of C. Let C = fx 2 C : x is a oint of density of Cg. Note tat jcnc j = 0. Let x 2 C. If is su ciently small, ji (x; ) \ Cj > =2 so te essential lim su of is. On te oter and, since for any > 0, te interval I (x; ) contains a rational number and ence a subinterval on wic = 0 so te essential lim inf of is 0. Tus as no limiting value at x and so all te more is not di erentiable tere. 2. does ave a zero L derivative for every ositive at almost every oint of C. Tis full measured subset of C will be a set of ositive measure and is te set romised in te statement of te teorem. Suose tat for eac > 0, is L di erentiable on C, a full-measured subset of C : Ten letting A = C nc, ja j = 0. Let A = [A n and C 2 = C na. Ten is not di erentiable on C 2, but is L di erentiable on C 2 for every > 0, since by de nition is L de di erentiable and

3 L DERIVATIVE 3 Holder s inequality imlies L di erentiability since de. Tus it is su cient to x and sow tat A as measure 0: On C we ave or, equivalently, (0.) Z Z j (x + t) (x) 0 tj dt! = = o () ; j (x + t) (x) 0 tj dt = o + ; as! 0. To sow tat ja j = 0, it su ces to sow tat for eac > 0, ja j <. Fix suc an and ick n so large tat (0.2) n > + and so large tat (n + ) 2 n+ < : Let B = [ n i= fx 2 C : dist (x; G i ) < 2 n g [ [ j>n x 2 C : dist (x; G j ) < 2 j. Ten jb j (2 2 n ) n + P j>n 2 2 j = (n + ) 2 n+ <, so it remains to sow tat (0.) olds for x 2 C nb so tat A B. Since x 2 C, (x) = and te absolute value of te left and side is ` = Z x+ x j (s) j ds = jc c \ Ij ; were I = [x ; x + ]. Assume < 2 n. Let G j be te rst comlementary interval tat meets I. Since x =2 B, j > n. Since 2 (i+)2 2 2 i2 and = 2, ` j[ ij G i j X ij 2 i2 2 2 j2 = 2 2 j j 2 j : Te last inequality olds because x =2 B imlies 2 j dist (x; G j ) and G j \ I (x; ) 6=? imlies dist (x; G j ). Since j > n > +, j is o + and relation (0.) follows. Tis examle slits ordinary di erentiation from all nite L di erentiation. Given any > 0, we can also create a function f for wic tere is a set E of ositive measure on wic f is di erentiable in te L q sense for every q < ; but f is not di erentiable at any oint of E in te L sense. We do tis by making a fat Cantor set te it stage comlementary oen intervals being centered at all (2j + ) =2 n and aving measure 2 i(+). Te details are sligtly more comlicated. Teorem 3 below does tis and a little bit more. Note tat te following teorem in articular searates te kt Peano derivative from all L kt Peano derivatives, 0 < <. Teorem 2. Tere is a set E of ositive Lebesgue measure and a function aving no limit at eac oint of E wic as a kt Peano derivative in te L sense for every natural number k and every ositive at eac oint of E. Proof. Te function and te subset of C of full measure aearing in te roof of te revious teorem are su cient for tis teorem also. In fact, for x 2 C set f 0 (x) = f (x) = for 2 (0; ); and set f i (x) = 0, for i = ; 2; : : : and

4 4 J. MARSHALL ASH 2 (0; ). Te de ning condition for aving a kt L Peano derivative at suc an x is Z! = f (x + t) 0t 0tk k! dt = o k or Z jf (x + t) j dt = o k+ : Te reasoning and calculations above remain uncanged, excet tat n must be cosen larger tan k + instead of larger tan +. Teorem 3. Let > 0 and k be a ositive integer. Tere is a set E of ositive Lebesgue measure and a bounded function nowere Peano di erentiable of order k in te L sense on E wic is Peano di erentiable of order k in te L q sense for every ositive q < at eac oint of E. Proof. Te case = and k = was treated rst. Ten followed te case = and general k: Te required examle for nite is te caracteristic function of a fat Cantor set wit te nt stage comlementary oen intervals being centered at all (2j + ) =2 n and aving measure c k 2 n(k+), were c k = 2 k. Te details follow. For N = ; 2; 3; : : : ; te comlementary intervals of rank N will be te oen intervals G in, i = ; 2; : : : ; 2 N, were te center of G in is centered at (2i ) =2 N and jg in j = c k 2 N(k+). Te center to center distance between contiguous intervals of rank N is 2 = 2 N. It will be convenient to work on [0; ] tougt 2 N of as a torus so tat in articular G N and G (2 N )N are contiguous. c, Let C = [ n= [ 2n i= in G = caracteristic function of C, x 2 C, and > 0. Note jcj = jc c j and jc c j P n= 2n c k 2 n(k+) = =2, so jcj > 0. Ten for any > 0, Z (x + t) (x) 0 t 0t2 0 tk Z (0.3) 2 k! dt = j (x + t) j dt = ji \ C c j ; were I = [x ; x + ]. Find m so tat 2 m < 2 m+. We ave for some j j, 2 m x < j +. Te comlementary interval G centered at te element of 2m j 2 m ; j + 2 m aving even numerator as rank at most m so tat te alf of G j interior to 2 m ; j + 2 m as measure at least c k 2. Tus 2 (k+)(m ) ji \ C c j c k+ k 2 2 m c k 2 k+ : We sow below tat wen q <, te rst k Peano L q derivatives of are 0 at a.e. x 2 C, so by Holder s inequality, if te L Peano derivatives exist at all, tey must be zero. However, combining tis inequality wit equation (0.3) sows tat Z (x + t) (x) 0 t 0t2 2! k! dt 0 tk! > ck 2 k

5 L DERIVATIVE 5 wic is not o k so does not ave a kt L Peano derivative at a.e. x 2 C. By te same reasoning as in te L case above, it is enoug to rove tat if q < are xed, and if > 0 is xed, ten tere is a set A = A (; q; ), A C suc tat jaj < and for every x 2 CnA, j[x ; x + ] \ C c j = o kq+ : (In te reduction to te su ciency of tis assertion, one needs to establis tis estimate directly for a countable set of q s tat belong to (0; ) and aroac.) Pick n suc tat 3 n <. Ten for eac ositive integer i, let A i be te oints of C wic are close to te comlementary intervals of rank i; exlicitly, for rank i, i n: let A i = [ 2i k= fx 2 C : dist(x; G ki) < n 2 g; and for rank j, j > n : let 2n A j = [ 2j k= fx 2 C : dist(x; G kj) < j 2 2 j g. Let A = [ i= A i, ten jaj nx ja i j + i= = 2 n 2 2 n X i=n+ nx j= 2 j = 2 n 2 2 n (2n ) + ja i j A + X i=n+ X i=n+ i 2 2 i 2 2 i 2i 2 Z n 2 + x 2 dx = 2 n 2 + n < 3 n < : n Let x 2 CnA and x > 0 so small tat < n 2. Let I = [x ; x + ]: Let 2n G be te rst comlementary interval intersecting I and let ` be te rank of G so tat jgj = c k. Note tat ` n + since is too small to allow any G of 2 (k+)` rank n to intersect I. Since G intersects I, (0.4) > `2 Let m = blog 2 c so tat 2 m < 2 m+ ; 2` : (0.5) m. log (=) : Let a (s) be te number of elements of rank s tat intersect I. Excluding te left-most and rigt-most elements, a (s) 2 centers of rank s intervals are in I and eac of te a (s) 3 distances between tese centers is 2 2, wence s (a (s) 3) 2 s+ 2, so (0.6) a (s) s : Since < 2 m, it follows tat (0.7) if s < m, ten a (s) 4:

6 6 J. MARSHALL ASH If ` < m, use inequalities (0.7) and (0.6) to obtain X ji \ C c j a (s) c k 2 (k+)s (0.8) s=` mx s=` X X 4 c k 2 (k+)s + 3 c k 2 (k+)s + c k 2 ks. 2 (k+)` + 2 km ; were A. B means tat for some constant C (k; ), A C (k; ) B. From tis and inequalities (0.4) and (0.5) we ave ji \ C c j. `2k+2 `22` s=m m 2k+2 k+ + k+. log 2k+2 (=) k+ = o kq+ : k+ k + 2 m s=m If ` m, te estimate is even simler; we get X ji \ C c j a (s) c k 2 (k+)s. k+ = o kq+ : s=m Acknowledgement. I tank David Preiss for very elful discussion. I also areciate te el of Alexander Stokolos. References [K] A. Kintcine, Recerces sur la structure des fonctions mesurables, Fund. Mat. 9(927), [CZ] A.-P. Calderón, and A. Zygmund, Local roerties of solutions of ellitic artial di erential equations, Studia Mat. 20(96), [LMZ] J. Luke¼s, J. Malý, and L. Zají¼cek, Fine toology metods in real analysis and otential teory. Lecture Notes in Matematics, 89, Sringer-Verlag, Berlin, 986. Deartment of Matematics, DePaul University, Cicago, IL USA address: mas@mat.deaul.edu URL: tt://condor.deaul.edu/~mas/