POROSITY OF CERTAIN SUBSETS OF LEBESGUE SPACES ON LOCALLY COMPACT GROUPS

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1 Bull. Aust. Math. Soc ), oi: /s POROSITY OF CERTAIN SUBSETS OF EBESUE SPACES ON OCAY COMPACT ROUPS I. AKBARBAU an S. MASOUDI Receive 14 June 2012; accepte 11 October 2012; first publishe online 12 December 2012) Abstract et be a locally compact group. In this paper, we show that if is a noniscrete locally compact group, p 0, 1) an q 0, + ], then { f, g) p ) q ) : f g is finite λ-a.e.} is a set of first category in p ) q ). We also show that if is a noniscrete locally compact group an p, q, r [1, + ] such that 1/p + 1/q > 1 + 1/r, then { f, g) p ) q ) : f g r )}, is a set of first category in p ) q ). Consequently, for p, q [1 + ) an r [1, + ] with 1/p + 1/q > 1 + 1/r, is iscrete if an only if p ) q ) r ); this answers a question raise by Saeki [ The p -conjecture an Young s inequality, Illinois J. Math ), ] Mathematics subject classification: primary 43A15; seconary 46E30, 54E52. Keywors an phrases: ebesgue space, σ-c-lower porous set, locally compact group, convolution. 1. Introuction an preliminaries Throughout this work, let enote a locally compact group with a fixe left aar measure λ. The moular function on the locally compact group is enote by. It is well known that is a continuous homomorphism on. Moreover, for every measurable subset A of, λa 1 ) = x 1 ) λx); A for more etails see [2] or [5]. For 1 p <, the ebesgue space p ) with respect to λ is efine as the Banach space of all equivalence classes of) Borel measurable functions f on with 1/p f p = f x) λx) <. If 0 < p < 1, it is known that p ) is a complete metric space with the metric f, g) = f g p λ f, g p )). c 2012 Australian Mathematical Publishing Association Inc /2012 $

2 114 I. Akbarbaglu an S. Maghsoui [2] In this case, for convenience, we put f p = f, 0) = f p λ. If p =, p ) is the Banach space of all equivalence classes of) essentially boune measurable functions f on with the norm f = esssup f. For measurable functions f an g on, the convolution f g)x) = f y)gy 1 x) λy) is efine at each point x for which the function y f y)gy 1 x) is aar integrable. For r [1, ], we write f g r ) to mean that f gx) < for λ-almost every x, f g is λ measurable on the set of all such x, an f g r <. Quek an Yap [6] prove the following interesting theorem. TEOREM 1.1. et be an infinite locally compact abelian group. et p, q > 1 be real numbers such that 1 < p <, 1 < q < an 1/p + 1/q > 1, an let r be efine by 1/r = 1/p + 1/q 1. Then: i) if is compact, then ii) iii) if is iscrete, then p ) q ) { s ) : r < s}; p ) q ) { s ) : s < r}; if is neither compact nor iscrete, then p ) q ) { s ) : s r}. Motivate by this result, Saeki [8] pose the following question. QUESTION. et be a locally compact group an let p, q, r [1, ]. If oes it follow that is iscrete? 1 r < 1 p + 1 q 1 an p ) q ) r ), Recently, łąb an Strobin [4], using the notion of porosity, generalise an consierably extene some interesting results on the convolution of functions essentially ue to Rickert [7] an Żelazko [10]. See also [1, 3, 8] for some relate results. et us recall the notion of porosity. et X be a metric space. The open ball with centre x X an raius r > 0 is enote by Bx, r). For a given number 0 < c 1, a subset M of X is calle c-lower porous if lim inf R 0+ γx, M, R) R c 2

3 [3] Porosity of certain subsets of ebesgue spaces 115 for all x M, where γx, M, R) = sup{r 0 : z X, Bz, r) Bx, R) \ M}. It is clear that M is c-lower porous if an only if x M, α 0, c/2), R 0 > 0, R 0, R 0 ), z X, Bz, αr) Bx, R) \ M. A set is calle σ-c-lower porous if it is a countable union of c-lower porous sets with the same constant c > 0. It is easy to see that a σ-c-lower porous set is meagre, an the notion of σ-porosity is stronger than that of meagreness. For more etails, see [4, 9]. In this work, we present a generalisation of the interesting result ue to Żelazko in [11]. We also answer the question aske by Saeki. 2. Results et us remark that we equip here the space p ) q ) with the complete metric efine by { max{ f1, f f 1, g 1 ), f 2, g 2 )) := 2 ), g 1, g 2 )} for p 0, 1), q 0, 1), max{ f 1, f 2 ), g 1 g 2 q } for p 0, 1), q [1, + ], for all f i, g i ) p ) q ) an i = 1, 2. We begin with the following theorem in which we use a technique from [4]. TEOREM 2.1. et be a noniscrete locally compact group an let p 0, 1) an q 0, + ]. Then for any symmetric compact neighbourhoo V of the ientity element of, the set E V = { f, g) p ) q ) : x V with f gx) < a.e.} is a σ-c-lower porous set for some c > 0. PROOF. et V be a symmetric compact neighbourhoo of the ientity element of. For a natural number n, put { } E n = f, g) p ) q ) : x V with f y) gy 1 x) λy) n a.e.. So, E V = n N E n. ence we only nee to show that for each n N, E n is c-lower porous for some c > 0. For this en we consier three cases. Case 1. p 0, 1) an q [1, + ). et sup x V x) = η an c 0, 1) be such that c 1 c + ηλv2 ) c = 1. λv) 1 c

4 116 I. Akbarbaglu an S. Maghsoui [4] Then, clearly, for 0 < α < c, α 1 α + ηλv2 ) α < 1. λv) 1 α By continuity of the map x α/x + ηλv 2 )/λv))α/x on 0, 1), we infer that there exist 0 < β < 1 α an > 1 such that ρ = 1 α β 1 ηλv2 ) λv) β 1) > 0. Fix a natural number n an suppose that f, g) E n. Since is not iscrete, inf{λu) : λu) > 0} = 0, an for R > 0, we can choose compact symmetric neighbourhoos an K containe in V such that K, λk)λv) λ)λv 2 ), f p λ < 1 α β)r an et s, t be such that λ) 1 1/p > n 2 β 1+1/p R 1+1/p η 1 1/p Define functions f an g on by setting 1 λv 2 ) ) 1/q ) 1 ρ. 2.1) sλ) = βr an tλk)) 1/q = βr. 2.2) { s x f x) := 1 )) 1/p if x, f x) otherwise an ence Moreover, gx) if x K, gx) := gx) + t if x K, Regx)) 0, gx) t if x K, Regx)) < 0. f f p = s 1/p x 1 ) 1/p f x) p λx) s x 1 ) λx) + f p λ s x 1 ) λx) + 1 α β)r βr + 1 α β)r = R αr. g g q = tχ K q = tλk)) 1/q = βr R αr.

5 [5] Porosity of certain subsets of ebesgue spaces 117 ence B f, g), αr) B f, g), R). It remains only to prove that B f, g), αr) E n =. Fix any h, k) B f, g), αr) an let A 1 = {x : hx) < 1 s x 1 )) 1/p }, A 2 = \ A 1, 2.3) an Then whence B 1 = {x K : kx) < 1 t}, B 2 = K \ B ) αr h f p hx) s 1/p x 1 ) 1/p p λx) A 1 1 s x 1 ) λx) A 1 1 = s λa 1 1 ) αr s 1 = s 1 x 1 ) λx) A 1 λa 1 1 ), 2.2) α ) = λ) β ) In the same way, by noting that gx) t, for x K, αr k g q k g)χ B1 q = t k t g t Noting that V, it follows that λb 1 ) R t 1) 2.2) = λk) β 1) ) χ B1 q t 1 λ) λv2 ) λv) ) λb 1 )) 1/q. β 1). 2.6) The above inequalities show that A 2 an B 2 are of positive measure an so nonempty. Now let z K be an arbitrary element, an efine the set F = A 1 2 z) B 2 an = zf 1. Since z K, A 1 2 z K. ence λ 1 ) = λfz 1 ) = λa 1 2 ) λa 1 2 \ B 2 z 1 )) λa 1 2 ) λk \ B 2)z 1 ) = λa 1 2 ) λb 1z 1 ) = λ) λa 1 1 ) z 1 )λb 1 ) 2.5),2.6) λ)ρ. 2.7)

6 118 I. Akbarbaglu an S. Maghsoui [6] Also, A 2, F B 2 an 1 z = F. Finally, we conclue that hy) ky 1 z) λy) 2.3),2.4) 2 s 1/p t y 1 ) 1/p λy) = 2 s 1/p t y 1 ) 1/p 1 y 1 ) λy) 2 s 1/p t η 1 1/p y 1 ) λy) = 2 s 1/p tη 1 1/p λ 1 ) 2.7) 2 s 1/p tη 1 1/p λ)ρ 2.2) = 2 β 1+1/p R 1+1/p 1 ) 1/q ρ η 1 1/p λ) 1 1/p λk) 2 β 1+1/p R 1+1/p 1 ) 1/q ρ η 1 1/p λ) 1 1/p λv 2 ) 2.1) > n. Case 2. p 0, 1) an 0 < q < 1. The proof is similar to the proof of Case 1 with q = 1. Case 3. p 0, 1) an q = +. et c = 1 2, so that c/1 c) = 1. Fix 0 < α < 1/2, so that α/1 α) < 1. By continuity of the map x α/x on 0, 1), there exist 0 < β < 1 α an > 1 such that α/β/ 1) < 1. Similarly to Case 1, we can choose a compact symmetric neighbourhoo containe in V such that f x) p λx) < 1 α β an λ) 1 1/p > nr 1+1/p 1 2α) 1 ρβ 1/p η 1 1/p ) 1, where sup x V x) = η an ρ = 1 α/β)/ 1). Define an { s x f x) = 1 )) 1/p if x, f x) otherwise, { gx) + R1 α) if Regx)) 0, gx) = gx) R1 α) if Regx)) < 0, in which sλ) = βr. By these efinitions, f f p < R αr an g g = R αr. ence B f, g), αr) B f, g), R). But B f, g), αr) E n =. To show this, take any h, k) B f, g), αr) an let 1 = {x : hx) > 1 s x 1 )) 1/p }, 2 = \ 1.

7 [7] Porosity of certain subsets of ebesgue spaces 119 Then α ) λ2 1 ) λ) an kx) R1 2α). β 1 Now let z be an arbitrary element. Define the sets F = 1 1z an E = zf 1. It follows that λe 1 ) = λfz 1 ) = λ1 1 ) = λ) λ 1 2 ) ρλ). Consequently, hy) ky 1 z) λy) R1 2α) 1 s 1/p η 1 1/p ρλ) E Thus h, k) E n, as require. = R 1+1/p) 1 2α) 1 ρβ 1/p η 1 1/p λ) 1 1/p > n. As an immeiate consequence of this theorem we obtain a result that generalises a well-known theorem of Żelazko [11] which states that p ), 0 < p < 1, is an algebra uner convolution if an only if is iscrete. COROARY 2.2. et be a locally compact group an let p 0, 1) an q 0, + ]. Then f g exists for all f p ) an g q ) if an only if is iscrete. PROOF. Recall that s ) t ) 1 ) if s t 1 an is iscrete. This proves the if part. For the converse we only nee to note that a σ-c-lower porous set is of first category, an p ) q ), ) is a complete metric space. TEOREM 2.3. et be a locally compact group an let p, q [1, + ) an r [1, + ]. If 1/p + 1/q > 1 + 1/r an is noniscrete, then for any symmetric compact neighbourhoo V of the ientity element of, the set is σ-c-lower porous for some c > 0. E V = { f, g) p ) q ) : f g r V, λ V )} PROOF. et V be a symmetric compact neighbourhoo of the ientity element of, an let p, q [1, + ) an r [1, + ] be such that 1/p + 1/q > 1 + 1/r. For a natural number n > 0, put E n = { f, g) p ) q ) : V r } f y) gy 1 x) λy)) λx) n ; if r = we instea consier the conition f y) gy 1 x) λy) n for λ-almost every x V in the above set. So, E V = n N E n. ence we only nee to show that for each n N, E n is c-lower porous for some c > 0. To prove this, let sup x V x) = η an c 0, 1) be such that c + η 1 c c 1 c λv 2 ) λv) = 1.

8 120 I. Akbarbaglu an S. Maghsoui [8] Then, clearly, for 0 < α < c, α + η 1 α α 1 α λv 2 ) λv) < 1. By continuity of the map x α/x + ηα/x λv 2 )/λv)) on 0, 1), we infer that there exist 0 < β < 1 α an > 1 such that ρ = 1 β 1) η λv 2 ) β 1) λv) > 0. Fix a natural number n an suppose that f, g) E n. Since is not iscrete, inf{λu) : λu) > 0} = 0, an for R > 0, we can choose compact symmetric neighbourhoos K an containe in V such that K, λ) < 1, an λk)λv) λ)λv 2 ) with 1/p 1/q f λ + g λ < 1 α β)r K an et s, t be such that λv) ) 1/q ) 1 λ) 1+1/r 1/p 1/q > n 2 β 2 R 2 η 1/p 1 ρ. λv 2 ) Define functions f an g on by setting sλ)) 1/p = βr an tλk)) 1/q = βr. { s x f x) = 1 ) 1/p if x, f x) otherwise, { t if x K, an gx) = gx) otherwise. Then B f, g), αr) B f, g), R). It remains only to prove that B f, g), αr) E n =. Fix any h, k) B f, g), αr) an let an Then an similarly 1 = {x : hx) < 1 s x 1 ) 1/p }, 2 = \ 1, R λb 1 ) t 1) B 1 = {x K : kx) < 1 t}, B 2 = K \ B 1. λ1 1 ) R = λ), s 1) β 1) = λk) β 1) λ) λv2 ) λv) β 1).

9 [9] Porosity of certain subsets of ebesgue spaces 121 Now let z K be an arbitrary element, an efine the set F = 2 1z) B 2 an = zf 1. Since z K, 2 1z K. ence λ 1 ) λ)ρ. Also, 2, F B 2 an 1 z = F. Finally, we conclue that hy) ky 1 z) λy) 2 st y 1 ) 1/p λy) = 2 st y 1 ) 1/p 1 y 1 ) λy) 2 st η 1/p 1 y 1 ) λy) Thus h, k) E n, as require. = 2 stη 1/p 1 λ 1 ) 2 stη 1/p 1 λ)ρ λv) ) 1/q 2 β 2 R 2 η 1/p 1 λ) 1 1/p 1/q) ρ λv 2 ) n > λ). 1/r The fact that a σ-c-lower porous set is of first category, together with Theorem 2.3, gives an answer to the question raise by Saeki. COROARY 2.4. et be a locally compact group an let p, q [1 + ) an r [1, + ] be such that 1/p + 1/q > 1 + 1/r. Then is iscrete if an only if p ) q ) r ). Acknowlegement The authors are very grateful to the anonymous referee for a very careful reaing of the paper an several valuable remarks that le to improvements. References [1] I. Akbarbaglu an S. Maghsoui, An answer to a question on the convolution of functions, Arch. Math. Basel) ), [2]. B. Follan, A First Course in Abstract armonic Analysis CRC Press, Boca Raton, F, 1995). [3] R. J. auet an J.. B. amlen, An elementary proof of part of a classical conjecture, Bull. Aust. Math. Soc ), [4] S. łąb an F. Strobin, Porosity an the p -conjecture, Arch. Math. Basel) ), [5] E. ewitt an K. Ross, Abstract armonic Analysis I Springer, New York, 1970). [6] T. S. Quek an. Y.. Yap, Sharpness of Young s inequality for convolution, Math. Scan ), [7] N. W. Rickert, Convolution of p functions, Proc. Amer. Math. Soc ), [8] S. Saeki, The p -conjecture an Young s inequality, Illinois J. Math ), [9]. Zajíček, On σ-porous sets in abstract spaces, Abstr. Appl. Anal ), [10] W. Żelazko, A note on p -algebras, Colloq. Math ), [11] W. Żelazko, A theorem on the iscrete groups an algebras p, Colloq. Math ),

10 122 I. Akbarbaglu an S. Maghsoui [10] I. AKBARBAU, Department of Mathematics, University of Zanjan, Zanjan , Iran S. MASOUDI, Department of Mathematics, University of Zanjan, Zanjan , Iran